I remember when early details emerged they had the factions already named and even made t-shirts and sold them. Then at release we have 3 factions and they don't really have a purpose.
An easy way to integrate factions with the storyline line is make it each ones objective to reclaim the lost moons and planets we once controlled. And YOU are the reclaimer helping unite their different goals by obviously going to each planet and help by killing.
Dead Orbit for instance would be committed to reclaiming the moon. Rep would be earned through patrols and strikes. Wearing the factions band would help for PVP where you compete against all of the other factions. They would have a character with cut scenes involved in the moon missions. Let's face it retaking earth alone is a hard task that's why each faction would be involved with the objective of taking their planet back in the common goal to reclaim our solar system.
Feed back appreciated.
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This would also allow room for more factions to emerge as the story expands farther out into the solar system.