Famous Finite Sets What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx 1{,}000{,}000$ elements.
I'll start the ball rolling with the five platonic solids. (Unfortunately not large.)
 A: As is well known, every finite (natural) number can be associated with a finite set of that cardinality. So in particular, the cardinality of a famous or special finite set must be a famous or special number. Here's a list of all the special numbers less than or equal to 9999 and contains quite a few items between $2^{10}$ and $2^{20}$.
Oh, what, you actually want the sets, and not just their cardinality, because there is more than one way of realising a set of a given cardinality? (Me grumbles something about bijective maps and isomorphisms of sets.) Fine:

*

*1132 is the number of 3-valent trees with 15 vertices

*1144 is the number of non-invertible knots with 12 crossings.

*1165 is the number of conjugacy classes in the automorphism group of the 12 dimensional hypercube.

*1205 is the number of fullerenes with 58 carbon atoms

*1294 is the number of 4 dimensional polytopes with 8 vertices.

*1378 is the number of symmetric idempotent 6×6 matrices over GF(2).

*1411 is the number of quasi-groups of order 5.

*...

*3240 is the number of 3×3×3 Rubik's cube positions that require exactly 3 moves to solve.

*3286 is the number of stable patterns with 16 cells in Conway's game of Life.

*...

*4535 is the number of unlabeled topologies with 7 elements.

*...

As beauty is in the eye of the beholder, I'm sure there are mathematicians out there who think each of the above numbers ought to be better known.
A: The sporadic groups? In particular they are finite sets... quite a few are too big to fit into your range, but the smallest (Mathieu groups) would do the trick.
http://en.wikipedia.org/wiki/Sporadic_groups
