  It is conventionally believed that a permutation test should ideally use all
permutations. If this is computationally unaffordable, it is believed one
should use the largest affordable Monte Carlo sample or (algebraic) subgroup of
permutations. We challenge this belief by showing we can sometimes obtain
dramatically more power by using a tiny subgroup. As the subgroup is tiny, this
simultaneously comes at a much lower computational cost. We exploit this to
improve the popular permutation-based Westfall & Young MaxT multiple testing
method. We study the relative efficiency in a Gaussian location model, and find
the largest gain in high dimensions.
