Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geomtric realization functor.

Is an analogous statement true for topological groups? That is, given a topological group $G$, regarded as a topological category with one object, does the geometric realization of its homotopy coherent nerve model the classifying space $BG$?