I want to build a finite CW complex such that $\pi_1$ is non-abelian and $H_i$ are zero for $i\geq 2.$ From Hatcher for a given group G, one can create an example of a 2-complex $X_G$ with $\pi_1(X_G)=G.$ I also checked from Mayer-Vietoris that if $G$ is cyclic such complex won't have any higher homology for $i\geq 2.$ I tried to take $G=S_3,$ the symmetric group of order 6 and from Mayer-Vietoris I get $H_2$ is $Z.$ I believe this was a correct calculation. Or is there a way to get the groups $G$, with $G$ non-abelian, such that we can get $\pi_1 =G$ and $H_i=0$ for $i\geq 2.$

Any reference or idea to create such an example? Or is there a way to claim such a finite complex can't exist?