Working on M11, I came up with that it can be generated using the following permutations:

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [[2, 0, 1, 7], [3, 4, 5, 6]] [[4, 0, 6, 7], [2, 3, 1, 5]] [[0, 7], [4, 6], [3, 5], [1, 2]]

The summary of that is that there's one big permutation which cycles everything, and a few more which allow GF(8) on the first 8 elements.

Do all the other sporadic groups have generators with a single cycle of everything and a few which work on the minimum number of other elements possible? If so, what are they?