Just out of interest, I've been checking Slimbridge's flamingo flocks on ISIS.
For the more common species, the numbers are pretty big.
ISIS lists (with animals listed under the <GROUP> section listed as unsexed):
Greater: 55.59.89 - total 203 birds
Caribbean: 44.45.54 - total 143 birds
Chilean: 8.7.105 - total 120 birds
Lesser: 16.9.24 - total 49 birds
Andean: 0.0.26 - total 26 birds
James': 2.0.0 - total 2 birds
So an overall total of 543 birds, with all six species represented. That's a good flamingo day out.
Interestingly, Slimbridge is currently the only zoo listing Andean and James' Flamingos on ISIS (as far as I'm aware both are still at Berlin and I saw a handful Andeans in the mixed flock at Wuppertal in May).
I think there are only very few institutions over the world that can compete with Slimbridge. Probably only Slimbridge and Berlin Zoo have a "full collection" of all 6 flamingo species. And for the sheer numbers - Singapore Bird Park, San Diego Sea World, La Palmyre, Bush Gardens Tampa, Cairo ... any others with over 300 birds?