I have done some "restricted" God's algorithm calculations for solving two layers of Rubik's Cube, given that the four edges of the face layer are already solved. I say it's a "restricted" calculation because it only considers certain moves or sequences of moves. For convenience, I will consider the two layers to be solved are the bottom layer (generally referred to as the D layer), and the layer above that (typically referred to by speedcubers as the E layer).

Given that four edges of the D layer are taken to be solved, the remaining four edges being considered (those that belong in the E layer) must be distributed among 8 possible edge locations. So the number of configurations for those four edges is 8*7*6*5*(2⁴) = 26880 (including orientations). The four corner cubies being considered (those that belong in the D layer) have 8*7*6*5*(3⁴) = 136080 configurations (including orientations). This makes for a total of 26880*136080 = 3,657,830,400 configurations.

I've been fooling around with symmetric Rubik cube patterns: "Cross" patterns, involving corners only; "Check" patterns, involving edges only; and "Dots" patterns, involving edges and corners.

By rotating the eight corners a quarter turn about one of the cube axes a cube state with a cross on four faces is produced. A second such state may be produced by rotating about a different cube axes:

R U B R U R D R U' D B' U' B' D' B' R' D' B'
R B D R B R F R B' F D' B' D' F' D' R' F' D'

By recursively forming binary products from these two generators a group of 24 cube states is produced composed of all the different pure cross patterns plus the identity element. This Cross group is composed of five conjugate classes:

Normally, we think of Rubik's cube symmetry in terms of a particular position x.  The symmetry of x can be defined as Symm(x), where Symm(x) is the set all symmetries m in M such that x^m=x, and where M is the group of 48 symmetries of the cube.  By the closure property, Symm(x) is a subgroup of M, and there are 98 possible such subgroups.  I wish to expand this definition of symmetry to encompass an entire subgroup of the Cube group G.

The primary motivation for the expanded definition is to provide a standard mechanism to deal with the symmetry of subgroups of the Cube group G such as H=<U,R>.  I'm probably going to make things more complicated than they need to be.  However, I don't recall seeing a general discussion of group symmetries before, neither on Cube-Lovers nor on this forum.

I have written an optimal solver console program for the quarter turn metric in C which compiles under Linux and Windows. The target subgroup for the pruning table is the permutations of <U,D,R2,F2,L2,B2> which have an even corner parity, lets call it H'. I experimented with another type of pruning table which does not store the distance to H' for each coset element but the moves which reduce the distance to H'. Because there are 12 possible moves in QTM, a 16bit word is enough for each coset element - a move reduces the distance to H' by one or increases it by one. The table fits in about 583 MB of RAM.