Coset complexes of $p$-subgroups in finite groups

Let $G$ be a finite group and $p$ be a prime. We denote by $C_p(G)$ the poset of all cosets of $p$-subgroups of $G$. We characterize the homotopy type of the geometric realization $|\Delta C_p(G)|$ for $p$-closed groups $G$, which is motivated by K.S.Brown's Question. We will further demonstrate that $\chi(C_{p}(G)) \equiv |G|_{p'} (\text{mod} p)$ for any finite group $G$ and any prime $p$.