There Can Be Only One

Make sure to go for Peacemaker to max out reputation with the factions, before turning your hand to destroying them all. As with Peacemaker this requires a map that has all five minor factions present - which you can check in the minor factions menu immediately after starting the game.

Also, it doesn't necessarily have to be you that destroys all the minor factions (contrary to the requirements for killing each individual one) - but you likely need to be the one that kills the last minor faction to be eliminated in the match (i.e. consistent with the individual-faction achievements). Hence this is soft-missable.

The factions are listed below, and there is an achievement for eradicating each one too:

1. Aluxian Resurgence - A Doomed Dynasty
2. Pirates - Pirate Hunter
3. Viturak Cabal - Get Off My Lawn
4. Pranas United - Breaching The Unbreachable
5. Jiskun Force - All Expeditions Must End