When are permutation invariants Cohen-Macaulay?

Over a field of characteristic 0, every ring of invariants of a finite group is Cohen-Macaulay. This is not true for fields of positive characteristic. We consider permutation representations and their invariant rings over fields ${\mathbb{F}}_p$ of prime order. We give an efficient algorithm which for any given permutation representation, determines those primes $p$ for which the invariant ring over ${\mathbb{F}}_p$ is Cohen-Macaulay, using linear algebra over $\text{ZZ}$. A generalization of the classical discriminant associated to the alternating group is defined for subgroups of certain finite unitary complex reflection groups.