I am looking for information related to the following question:
For which $n \in\mathbb{N}$ can every group $G$ be realized as $\pi_n(X)$ for some space $X$?
I have seen in Hatcher that $n=1$ is one such case. I was wondering whether this result was true for higher $n$ as well. I tried a google search, but I didn't turn up anything, perhaps because I didn't phrase the question in an optimal manner. Is there any information out there related to this question?