Show that there is a nonabelian group of order $p^n$ with a cyclic subgroup of index $p$

Question

Assume $p$ is a prime integer and let $n$ be an integer such that $n \geq 3$.

Show that there is a nonabelian group of order $p^n$ with a cyclic subgroup of index $p$.

Answer 1

The dihedral group $D_8$ is of order $8=2^3$ and it is non-abelian group. Also, any subgroup of $D_8$ with order $2$ is cyclic.