Well Omicron Alpha is the main home system for the outcasts but I would warn you before hand that the outcasts are no easy opponents and are probably one of the best fighter factions in the game.

The Dark One,

Dark Sovereign

To get In with any Crime faction just blast Bounty hunters and the local police/navy. But kill just enough to get them neutral unless you want the police etc after you then just kill all of them. Easier and quicker than killing Outcasts to make the other factions friendly.

Stuff the bounty hunters nobody is allied with them so make them your enemies they always try to take your cargo and when you kill them your reps with the rebel causes usually goes up so go for it kill the bounties thats my advice also go for Red Hessians.

once i tried killing outcasts to get the hessian and corsair rep up, but there was no effect, go with killing police and bh. Bh do have good fighters tho so dont get em hostile.

you cant get both corsairs and hessians to like you without modding. If you can, I'll kick myself