  We consider a variant of random walks on finite groups. At each step, we
choose an element from a set of generators ("directions") uniformly, and an
integer from a power law ("speed") distribution associated with the chosen
direction. We show that if the finite group is nilpotent, the mixing time of
this walk is of the same order of magnitude as the diameter of a suitable
pseudo-metric, D(S, a), on the group, which depends only on the generators and
speeds. Additionally, we give sharp bounds on the $\ell^2$-distance between the
distribution of the position of the walker and the stationary distribution, and
compute D(S,a) for some examples.
