Identifying topological spaces whose fundamental groups are $\mathbb{Z}/n\mathbb{Z}$

Question

For any $n \in \mathbb{N}$ I want to compute all path connected and locally path connected topological spaces $X$ (up to homotopy equivalence) whose fundamental group is $\mathbb{Z}/n\mathbb{Z}$. Actually for any prime $p$ I know the construction of the lens space with fundamental group $\mathbb{Z}/p\mathbb{Z}$. But are there any other spaces with $\mathbb{Z}/p\mathbb{Z}$ as it's fundamental group (which are not homotopy equivalent)? And what about any natural number $n$ instead of only primes $p$?

Thanks in advance.

Answer 1

If $\pi_1(X) = \mathbb{Z}/n\mathbb{Z}$, then for any simply connected space $Y$, $\pi_1(X\times Y) = \pi_1(X) = \mathbb{Z}/n\mathbb{Z}$.

Answer 2

Suppose $X$ is a path connected and, say, locally contractible space with fundamental group $G$. Then it has a universal cover $\widetilde{X}$ on which $G$ acts freely such that $X$ is the quotient $\widetilde{X}/G$. Up to homotopy equivalence, every simply connected space can occur as $\widetilde{X}$, and it turns out that the classification of possible choices of $X$ is equivalent to the classification of pairs consisting of a simply connected space and an action of $G$ on it (in a suitably homotopy-theoretic sense).