 Solution

The polygon presentation of the Klein bottle, or Seifert-van Kampen theorem applied to its cell structure, givesSend $a$ to $(12)$ and $b$ to $(123)$ in $S_3$. Conjugation by $(12)$ inverts $(123)$, so this defines a surjective group homomorphism. Since $S_3$ is nonabelian, the fundamental group is nonabelian.