# Subgroup enumeration

Submitted by B MacKenzie on Sun, 04/12/2009 - 15:07.

I've been playing around with the Rubik's cube subgroup generated by the turns: (U U' D D' L2 R2 F2 B2) which I refer to as the D4h cube subgroup after the symmetry invariance of the generator set. This, I believe, is the subgroup employed by Kociemba in his two phase algorithm. Anyway, I have performed a partial enumeration of the subgroup and its coset space. I was wondering if anyone might be able to confirm these numbers as a check on my methodology.

Six Face/D4h Coset Enumeration (q turns) Depth Cosets Total 0 1 1 1 4 5 2 34 39 3 312 351 4 2772 3123 5 24996 28119 6 225949 254068 7 2017078 2271146 D4h cube group enumeration, 8 gen (U U' D D' L2 R2 F2 B2) Depth Class(D4h) Elements Total 0 1 1 1 1 2 8 9 2 7 48 57 3 25 284 341 4 124 1678 2019 5 648 9664 11683 6 3523 54475 66158 7 19006 299960 366118 8 100741 1602352 1968470 9 518843 8279732 10248202 10 2571647 41089158 51337360 D4h cube group enumeration, 10 gen (U U' D D' L2 R2 F2 B2 U2 D2) Depth Class(D4h) Elements Total 0 1 1 1 1 3 10 11 2 12 73 84 3 48 514 598 4 262 3515 4113 5 1550 22984 27097 6 9245 142982 170079 7 54578 859946 1030025 8 309714 4922028 5952053 9 1681312 26815882 32767935