The Galaxy Police is an intergalactic law enforcement agency. Their duties pertain to giving criminals danger levels, as well as capturing said criminals.
Head of the GP - Mikami Kuramitsu (also Head of the GP Academy)
Head of Docking Maintence - Minami Kuramitsu (formerly Grand Marshal)
The Galaxy Police seems to possess highly-advanced technology, dedicated to the capture of criminals. A notable example would be the Control Cube, used perhaps most prominently by Mihoshi in the original series.
Operated by turning its sections much like a Rubiks cube, the Control Cube's functions, at least in the first continuity, seem to revolve around small warps in space, apparently to easily access equipment and ships. In other continuities (like Tenchi Universe), it doesn't get quite as much use as a vital tool, and has slightly different uses, including a tracking system.
Another example would be Mihoshi's ship, once again in the original series. According to some sources, that ship possesses an experimental design and the ability to evolve itself over time.
The Galaxy Police has connections to Jurai, as the powerful Kuramitsu family, which has several members in the Galaxy Police's ranks, has a close relationship with the Jurai Royal Family. This element remains in the Tenchi Universe and Tokyo continuities, but is left more ambiguous than in the original series and Tenchi Muyo! GXP, as the bureaucracy is not as widely explored in these series and Mihoshi's family does not appear in them. In fact, throughout the second half of Universe, Tenchi and his family were actually chased by the Galaxy Police thanks to Kagato's coup of Planet Jurai under the name of Yosho vilifying them as criminals.
Every member must go through the body enhancement during the training to better perform their duties, boosting strength and endurance to several times normal human level.
Tenchi Muyo! GXP
In this spinoff continuation of the OVA series, the Galaxy Police wanted to recruit Tenchi (because Amane had contacted Tenchi, through her friend and colleague Mihoshi, about his entering the G.P. Academy), but got Tenchi's close friend Seina Yamada, a young man with such terrible luck his only friends were Masakis of some branch or another, by mistake. GP was disappointed that they didn't get Tenchi and Tenchi wasn't interested in joining GP. Second Class GP detective Amane Kaunaq comes to Earth looking for Tenchi to give him a pamphlet and application to join the GP, but typical of Seina's luck, Amane's ship crash-landed at the lake near the Masaki residence, causing a wave and washing out Seina, who was visiting at the time. When he woke up he was lying on the dock of the Masaki home saved by Amane, who mistook him for Tenchi's pupil due to their similar appearance and offered Seina the application to instead.
Believing that there was a prize involved, Seina's family forcibly nabbed his signature and thumb print to fill out the application in order to get him to join. When he woke from sleep for the night, he found himself aboard a spaceship, recruited to be the newest member of the G.P. Academy. He met Mitoto Kuramitsu, the ditzy mother of Mihoshi and cleaning lady extraordinaire, before discovering his childhood friend Kiriko Masaki was also a GP officer. Amane and Kiriko were demoted by their superiors and were order to train Seina. As it was against protocol to accept members of underdeveloped planets as officers, Seina's being accepted into the GP was not taken kindly at first. However, Seina's luck, and the favor he found from the "Devil Princess" Seto Kamiki Jurai as a result, saw him thrown into a plan to stop the Daluma pirate guild from taking over the galaxy, as well as in command of a GP fleet including Amane, Kiriko, two-millennia-old child priestess Neju Na Melmas, and the pirate Ryoko Balta who reformed after several encounters with him in space.
The Kuramitsu Family
The Kuramitsu Family are dominant in the Galaxy Police due to the fact that several of the G.P forces are actuallly derived from Seniwa, where the Kuramitsu Famly is dominant. It is shown that Mikami Kuramitsu, is, in fact, the true power behind the Galaxy Police, thanks to her status as the head of the family. She shows this as she is able to remove Minami as Grand Marshal.
GP stated that people from underdeveloped planets can't be accepted to the G.P. Seina was accepted by accident. However, there are ways to be accepted into the G.P. only if your worthy.
Over time, Senia became a great member of the G.P.
G.P. tried to recruit Tenchi because Tenchi was half Jurai despite coming from Earth, underdeveloped planet.
Every member must go through the body enhancement during the training possibly to be able to keep up with the Juraians physical skills.