Algorithm for factoring elements of permutation groups?

You can solve a Rubik's cube by factoring its permutation into a sequence of "elementary" permutations (a subset of permutations that is sufficient to construct every other permutation in the group). There are several algorithms for solving Rubik's cubes. Are there any algorithms for the general problem of factoring permutations?

It seems that your problem could be stated in terms of finding paths in Cayley Graphs, which might interest you:

Cayley Graph - Wikipedia

StackExchange Computer Science Q#109708, answer score: 2