Abstract: I will show that the universal cover of the commutative classifying space has the homotopy type of a wedge of spheres when $G$ is an extraspecial $p$-group. Commutative classifying space is a certain subspace of the usual classifying space whose set of $n$-simplices is given by the set of commuting $n$-tuples in $G$. Its universal cover can be described as a partially ordered set obtained from the collection of abelian subgroups of $G$. Such objects are closely related to Tits buildings associated to algebraic groups. I will also mention about possible applications of such objects in quantum computation.