Yes, if $G$ is an abelian group, its classifying space $BG$ is an *abelian* topological group whose $\pi _1$ is $G$. You can find details in Joan Baez's  wonderful post,  hearteningly  called "Classifying Spaces Made Easy"

http://math.ucr.edu/home/baez/calgary/BG.html    

or in the answer by Chris Schommer-Priess to the following question on this site:

http://mathoverflow.net/questions/12144/classifying-space-of-a-group-extension