I have used GAP to compute the subgroup generated by 300 random pairs of
elements of G. 151 of those pairs generated the entire group, so the
probability is about 50%.

I don't think we can figure out the exact number, since we don't know the
maximal subgroups of G. One maximal subgroup we know is the derived
subgroup (on which the upper bound of 75% is based). Then there are the
8 stabilizers of the corners (of index 8) and the 12 stabilizers of the
edges (of index 12). Using those it should be possible to push the upper
bound down to something about 60%.

Martin.

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Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,   +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany