The minimal permutation degree of a finite group G is the smallest non-negative integer n such that G embeds inside Sym(n). This invariant is easy to define but very difficult to calculate in general. Moreover, it doesn't behave well under algebraic constructions such as direct product and homomorphic image. For example, it is possible for the minimal degree of a homomorphic image to be strictly larger than that of the group – such groups are called 'exceptional'.

In this talk, I will describe how this invariant maybe calculated by a greedy algorithm for nilpotent groups and report on recent work with Britnell and Skyner on classifying exceptional groups of order p^5.