Enumerating classes and characters of 𝑝-groups

Abstract

We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p p -groups of nilpotency class less than p p . This allows us to unify and generalize a number of existing enumerative results, and to obtain new such results for generalizations of relatively free p p -groups of exponent  p p . Our main tools are the Lazard correspondence and the Kirillov orbit method.