On the homotopy type of the non-completed classifying space of a p-local finite group

We establish sufficient conditions for the nerve of the centric linking system of a p-local finite group (S, , ) to have the homotopy type of an Eilenberg-MacLane space K(G, 1) for a group G which contains S. We prove that in this situation the entire p-local finite group can be reconstructed from G. Our theorem applies to many of the known examples of exotic p-local finite groups.