$K(Z,2)$ out of circular permutations

We discuss ${SC}_{\ast }$, a simplicial homotopy model of $K(Z,2)$ constructed from circular permutations. In any dimension, the number of simplices in the model is finite. The complex ${SC}_{\ast }$ naturally manifests as a simplicial set representing ``minimally" triangulated circle bundles over simplicial bases. On the other hand, existence of the homotopy equivalence $\mathrm{\mid }{SC}_{\ast }\mathrm{\mid }\approx B(U(1))\approx K(Z,2)$ appears to be a canonical fact from the foundations of the theory of crossed simplicial groups.