Is there a classification of finite groups $G$ with the property that any prime dividing $|G|$ must also divide $|G^{ab}|$ (the order of its abelianization)?

Being nilpotent is sufficient, though not necessary. On the other hand, being supersolvable is not sufficient, since the abelianization of any dihedral group $D_{2k}$ is a 2-group, and is also not necessary.

Are there other interesting families of finite groups which have this property?

(In particular I'm interested in groups generated by 2 elements)