A classification of the stable type of $BG$

We give a classification of the p-local stable homotopy type of BG, where G is a finite group, in purely algebraic terms. BG is determined by conjugacy classes of homomorphisms from p-groups into G. This classification greatly simplifies if G has a normal Sylow p-subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow p-subgroup. If G is cyclic ${\bmod \;p}$ then BG determines G up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group BG can be written as a unique linear combination of BH's, where H is cyclic ${\bmod \;p}$.