For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not products of Eilenberg-MacLane spaces. This rules out obvious constructions using crossed modules (at least, obvious to me).

One idea is this: take a representation $\rho:\Gamma_g \to SO(3)$ of $\Gamma_g = \pi_1(\Sigma_g)$, the fundamental group of a compact, connected, orientable surface or genus $g$. Then form the associated sphere bundle $X=\widetilde{\Sigma_g} \times_\rho S^2 \to \Sigma_g$.

Then, unless my calculation is wrong, $X$ is a ~~2-type~~ space with $\pi_2(X) = \mathbb{Z}$, $\pi_1(X) = \Gamma_g$ and $k$-invariant $a\in H^3(\Gamma_g,\mathbb{Z})$.

So my question is,

are there any non-trivial representations $\rho$ or cohomology classes $a$?

and secondarily,

would a non-trivial representation $\rho$ give rise to a non-trivial $k$-invariant in the above situation?