I now have a proof that no element of the cube group can be of order
greater than 1260. Since you have so thoughfully provided me with an
element of order 1260, I must conclude that this element is indeed the
maximum, as you claimed (but where did the number 216=6^3 come from?).

My proof contains no nice derivation of the number 1260, you will be
dissapointed to see where it comes from, it is just all that is left
after a number of cases have been eliminated.

Perhaps someone can devise a "nice" proof of this fact. Is this fact
in the literature? (Bernie?)