Let $G$ be a compact Lie group and let $\mathcal{P}_G$ denote the family of proper subgroups of $G$. The universal space for the family $\mathcal{P}_G$ is a cofibrant $G$-space which does not have $G$-fixed points and such that for every proper subgroup $H<G$, the fixed point space $(E\mathcal{P}_G)^H$ is contractible. These properties determine $E\mathcal{P}_G$ uniquely up to $G$-homotopy equivalence.

I am particularly interested in the case where $G$ is a finite cyclic group. When the order of $G$ is prime, I know that I can take the model $E\mathcal{P}_G=EG$, the non-equivariant universal space associated to the group $G$. What I would like to know is if there are concrete models for $E\mathcal{P}_G$ for general cyclic groups. My end goal is to compute the mod-2 singular homology of such spaces, but I find it difficult to do so with just the properties characterising such spaces.

In general, I am also interested in knowing how one goes about constructing $E\mathcal{P}_G$ for general (compact Lie) groups.