Topological group with nonzero Euler characteristic

Question

Does there exist a compact, Hausdorff and path-connected topological group $G \neq \{*\}$ whose cohomology $H^*(G;\mathbb{Z})$ is of finite type, and the Euler characteristic $$ \chi(G) := \sum_{i=0}^{\infty} (-1)^i rank(H^i(G;\mathbb{Z}))$$ is nonzero? Note that such a $G$ cannot be triangulated, and cannot even be an ANR because of Lefschez fixed point theorem.