Experimenting with finite groups recently, I’ve learnt a few things I was completely unaware of:
(1) there are so many distinct groups of order 512 (or 1024, 1152, 1536 and 1920) up to isomorphism, that Magma and GAP are not able to recognize them (they have databases of groups of order up to 2000, except for 1024, but given one abstract group of this order, they can not pinpoint which element of the database it is). Precisely, there are 10494213 distinct finite groups of order 512 up to isomorphism, 157877 of order 1152, 408641062 of order 1536 and 241004 of order 1920. These numbers are several order of magnitude larger than what I would have guessed if asked point-blank.
(2) the order of groups of a fixed Lie type, say X(q), where X could be PSL, is not always monotonic with respect to the variable q; for instance:
gap. Order(PSL(3,17));
6950204928
gap. Order(PSL(3,19));
5644682640
gap. Order(PSL(3,29));
499631102880
gap. Order(PSL(3,31));
283991644800
(Of course, this is then easy to check, and the point is that when q-1 is divisible by 3, the order drops as the center of SL(3,q) becomes a bit bigger).
