Someone on the Yahoo forum asked about how to do a 2x2x2 God's algorithm calculation and mentioned the "1152-fold" symmetry for the 2x2x2. I got to looking at some of the messages in the Cube-Lovers archives that Jerry Bryan had made about B-conjugation and the 1152-fold symmetry of the 2x2x2. He found that there were 77802 equivalence classes for the 2x2x2.

I have used antisymmetry to further reduce the number of equivalence classes for the 2x2x2 to 40296. The following table shows the class sizes of these equivalence classes.

class size  class size/24  count
----------  -------------  -----
     24           1            1
     48           2            1
     72           3            3
     96           4            1
    144           6           14
    192           8           11
    288          12           49
    384          16           22
    576          24          337
    768          32            6
   1152          48         3353
   2304          96        36498
                           -----
                  total    40296

I then performed God's algorithm calculations (HTM and QTM) to find the number of equivalence classes at each distance from the solved 2x2x2 cube. The results are given below. Because the 2x2x2 has no centers that provide a reference for the positions of the other cubies, the number of positions of the corners for the 2x2x2 (the only cubies it has) can be considered to be 1/24 the number of positions of the corners of the 3x3x3 (3674160 instead of 88179840). So in the tables below, I use the factor-of-24 reduced numbers for simplicity. The tables further break down the positions with respect to different class sizes.

The method I used was suggested by Tom and Silviu's coset searches for the Rubik's Cube: Starting from a cube-shaped, odd-parity position of Square-1, an iterated depth-first search was made for all even-parity cube-shaped positions, with the search being pruned on [shape]x[parity].

           
|x| Symmetry  Size           Patterns        Positions
     Class     of
                xM


  0      M       1               1               1
      Total                      1               1
      
  1      CR     12               1              12
      Total                      1              12
      
  2      I      48               2              96
         Q       6               1               6

           
|x| Symmetry  Size           Patterns        Positions
     Class     of
                xM

  0      M       1               1               1
      Total                      1               1
      
  1      CR     12               1              12
         Q       6               1               6
      Total                      2              18
      
  2      I      48               4             192

I have received permission to post Dan Hoey's taxonomy of symmetry groups of Rubik's Cube.  Also, I will relate Dan's taxonomy to Shoenflies symbols as implemented in Herbert Kociemba's Cube Explorer.  (Go to http://kociemba.org/cube.htm and then click on Symmetric Patterns.)  To that end, some preparatory comments are in order.

In order to define any terminology for the symmetry groups of Rubik's Cube, it's necessary first to define some terminology for the symmetries of the cube.  To the best of my knowledge, no standard terminology has been adopted by the Rubik's cube community for the symmetries of the cube.  The terminology I'm going to use is very similar to some terminology I have seen before, but I can't remember the reference.  It may have been Christoph Bandelow's book, Inside Rubik's Cube and Beyond.  In any case, if I can find the reference I want to give proper credit.

I have completed a five-stage analysis of the 4x4x4 cube showing that it can always be solved using at most 68 turns. The analysis used the same five stages that were used in my prior posts where I claimed the 4x4x4 cube can be solved in 79 single-slice turns, or alternatively in 85 twists. The difference in this analysis is that it allows any single layer turn or double layer turn (where the two layers are any two adjacent layers and moved together) to be counted as a "turn." In some prior posts, I referred to these turns as "block turns." So the set of turns about the U-D axis that count as one turn are the following:
  U,U',U²,u,u',u²,(Uu),(Uu)',(Uu)²,
  D,D',D²,d,d',d²,(Dd),(Dd)',(Dd)²,
  (ud'),(u'd),(ud')²