We develop fast parallel solutions to a number of basic problems involving solvable and nilpotent permutation groups. Testing solvability is in NC, and RNC includes, for solvable groups, finding order, testing membership, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in RNC, find the center, a central composition series, and point-wise stabilizers of sets. There are applications to graph isomorphism. In fact, we exhibit a class of vertex-colored graphs for which determining isomorphism is NC-equivalent to computing ranks of matrices Over small fields. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to RNC.