TITLE: Orbits of independent sets of the hypercube
QUESTION [5 upvotes]: How does one enumerate the distinct orbit classes of independent sets of the hypercube modulo symmetries of the hypercubes? 
The counting of the number of independent sets in an $n$-dimensional hypercube modulo symmetries of the hypercube has been done up to $n=5$ by D.Eppstein as seen in the OEIS. We do have a bound on the number of independent sets of a regular graph as found by Y.Zhao. But I'm not aware of other results to this old problem of mine.
Does anyone know of a resource on how to go about this enumeration? I've already coded a brute-force program to do this listing and had reproduced Eppstein's results up to $n=5$. The $n=6$ took too much time and memory space on my personal computer that it always crashed. But this was years ago. I was just reminded of this recently, so here's my first post on MO.

REPLY [6 votes]: This is an easy problem for an orderly algorithm, which can generate the independent sets without isomorphs with no direct isomorphism checks between different independent sets.
It takes a couple of minutes on a recent computer and concludes that there are there are 519195 independent sets in the 6-cube up to isomorphism.
The numbers of each size (from 0 to 32) are:
1, 1, 5, 11, 56, 182, 742, 2323, 6714, 15233, 29200, 44870, 58711, 64916, 64902, 58982, 51064, 40835, 30884, 21145, 13511, 7631, 4058, 1859, 839, 316, 133, 42, 19, 5, 3, 1, 1