Well, the largest single corporation is Eve University, who have 1500 members. That's a big step up from the next largest though, which is nowhere near as big.
Goonswarm are the largest alliance, and I think they had nearly 4000 members at one point, though after the problems they had when one of the alliance directors turned traitor and collapsed the entire alliance, I imagine some of their member corps would have moved on. The reformed GS alliance is still well over 3000 members though.
I think BoB were bigger still, back in their heyday, before they were destroyed.
Anyway, GoonFleet (or whatever the new Goon corp is called), who are the main corp in the GoonSwarm alliance, works on a similar model to PCG - only members of the Something Awful forums (who have been active for at least 3 months, and didn't just join in order to gain access to the corp) are allowed to join - and that corp alone is a good 800 members or so. The other corps are non-goons, but there are still pretty strict entry criteria to join the alliance, IIRC.
So, you never know - one day in the distant future, we may be among the most influential groups in Eve, holding a large area of space, and commanding several Titans. It will take a loooong time though.