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I just found a recent post by "secondmouse" on sci.math that deserves a wider audience. I'll quote it here in full.

I found the following short presentation for the
miniature 2x2x2 Rubik's cube of order 3674160:

    < a,b,c | a^4 = b^4 = c^4 = 1,
                 ababa = babab,
                 bcbcb = cbcbc,
                 abcba = bcbac,
                 bcacb = cacba,
                 cabac = abacb,
                 (ac)^2 (ab)^3 (cb)^2 = 1 >

See the following link for more info as to why