Let n≥3. Then the dihedral group on n vertices, D2n, is not abelian.
The most natural dihedral group presentation is ⟨a,b∣an,b2,bab−1=a−1⟩. In particular, ba=a−1b=a−2ab, so ab=ba if and only if a2 is the identity. But a is the rotation which has order n>2, so ab cannot be equal to ba.