Brown complex

Definition
Given a finite group $$G$$, the Brown complex of $$G$$ is defined as follows. Let $$p$$ be a prime dividing the order of $$G$$ and denote by $$S_p(G)$$ the set of nontrivial $$p$$-subgroups of $$G$$. The order complex associated with $$S_p(G)$$ is termed the Brown complex for $$G$$.

More explicitly, the Brown complex is a complex where the points are $$p$$-subgroups and an $$r$$-simplex is a set of $$p$$-subgroups of which any two are comparable.