Singmaster states that the diameter of the group for the 2x2x2 cube is not known. I do not know whether it has been calculated in the mean time, so I just did calculate it. The number of elements in the group is 3,674,160. (Fix one corner, the others allow every permutation and one third of all possible twists.) The diameter is 11 if we do allow half-turns, it is 14 if we do not allow half-turns. The distribution is: If we allow half-turns: 1 with 0 moves 9 with 1 moves 54 with 2 moves 321 with 3 moves 1847 with 4 moves 9992 with 5 moves 50136 with 6 moves 227536 with 7 moves 870072 with 8 moves 1887748 with 9 moves 623800 with 10 moves 2644 with 11 moves If we do not allow half-turns: 1 with 0 moves 6 with 1 moves 27 with 2 moves 120 with 3 moves 534 with 4 moves 2256 with 5 moves 8969 with 6 moves 33058 with 7 moves 114149 with 8 moves 360508 with 9 moves 930588 with 10 moves 1350852 with 11 moves 782536 with 12 moves 90280 with 13 moves 276 with 14 moves In the first case heuristics give a diameter of at least 9. We see that the majority of the configuration is within distance 9 from start. So it appears that heuristics get close to the real value. We see also that in both cases there is more than one diametrally opposite configuration. Next I will find out which those are (and if they have something in common).
BTW, calculation did not take very long, only a few (<3) minutes on an FPS
(i.e. an extremely fast SPARC). But as the calculations are memory bound
rather than compute bound, the speed of the processor is not so very important.