The Galactic Space Police Brotherhood is an extraterrestrial organization. Very little is known about this organization other than Haruko claiming that she's a 1st Class Space Patrol Officer, the organization is given by name in the manga, but not in the anime. It is currently unknown if this organization is a government, a society or just a unified military force; though it is likely at the very least the latter as it's implied that they are at war with Medical Mechanica and they have an apparent military ranking system.
Haruko is sometimes seen talking the Miyu Miyu, referring to Naota as useless when he's N.O. channel doesn't immediately yield results and is later seen, in episode 5, trying to convince someone that she doesn't have feelings for Atomsk. She's presumably using the cat to communicate with her superiors; it's possible that this works by using the cat's brain to transfer sound in the same way that Naota's N.O. channel can transfer matter.
In the Anime the only real presence of the Galactic Space Police Brotherhood is Haruko's communications with them through Miyu Miyu, however in the manga Haruko states that they are "the good guys", but they order an airstrike on Mabase in order to strike at Medical Mechanica with disregard for the city's inhabitants. In the anime the Galactic Space Police Brotherhood is more ambiguous, while in the manga they seem malicious or just generally unconcerned of the safety of bystanders.