Do you have any advice for doing that? It sounds like a really cool idea!
I recommend just taking whatever the mainstream faction idea would be and making that your topic, like a thieves guild or a religion, and then seeing how those ideas morph.
If the factions are all very different types of groups (one's a religion, one's a merchant cartel), play multiple times, once for each type of faction.
Ben nailed it already, so I'll just kind of post this as a longform agreement. We just defined the game as about cults (they're a core part of the campaign we're playing). I'm not sure how far I'd push my luck with trying to create different kinds of stuff, but if I ever needed to say, generate a host of new gangs for Blades in the Dark, or maybe immortal beings for a Lord of the Rings esc fantasy, etc etc, then I'd reach for this game again.
I'd also just play it as intended too, which I'm actually yet to do. It sits in a really nice place for prompting and easy creativity.
EDIT: In re-reading this, I've realised that I implied something without saying it. We had concrete truths as we'd been playing in the setting for a bit already. Magic exists. This is the normal technology level. Blah blah blah. Then everything above that layer is fair game as usual. We also all knew the intention of this exercise going in.