It has been a longstanding goal of group theory to somehow describe if not characterize finite p-groups. In the 1940's, P. Hall proposed to tackle the problem by considering groups only up to their commutator structure, a suggestion that has turned out to be quite efficient. Many authors have since studied the relations between commutators, in particular some universal ones via the exterior product and more recently the Bogomolov multiplier. The commutators in a finite group may however well be related in a non-universal manner. We will take a look at the building blocks of these groups, i.e. minimal groups possessing non-universal relations, and show how their restricted structure enables a probabilistic study of the universality of commutator relations.