Is this 2-complex a $K(\pi,1)$? Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, so we have $\widetilde H_i(X)=0$ for all $i$. And if $\pi_2(X)=0$, we have that $X$ is $K(\pi,1)$.
My question is: how can one compute $\pi_2(X)$? Computing homotopy groups is hard, what methods may i use?

Letting $\widetilde X$ be the universal cover of $X$ we have
$$\pi_2(X) \approx \pi_2(\widetilde X) \approx H_2(\widetilde X,\mathbb{Z})
$$
The first isomorphism comes from the long exact sequence of homotopy groups of a fibration, using discreteness of the fiber of the universal covering map. The second isomorphism comes from the Hurewicz theorem, using simple connectivity of $\widetilde X$.
