Some Explorations in Visual Group Theory

Looking at a table of the orders of small finite groups, that there are 2328 of order 128 and 8681 of order 576 struck me: Why don't we make videso showing all of their Cayley Tables.

Bounty: I am offering a bottle of whisky to anyone who can successfully translate Jacob Lurie's Higher Topos Theory into analytic combinatorical language in the manner of Flajolet's Analytic Combinatorics.