An arithmetic theorem related to groups of bounded nilpotency class

We characterize the set of positive integers m having the property that every group of order m is nilpotent of class at most c, where c is a fixed positive integer or infinity. This generalizes and relates results of Dickson and Pazderski. The special case where c=1 (all groups of order m are abelian) is used to construct a substantial class of finite Schreier systems S in free groups such that S is not a right transversal for any normal subgroup.