Groups of prime-power order with a small second derived quotient

For odd primes we prove some structure theorems for finite $p$-groups $G$, such that $G''\neq 1$ and $|G'/G''|=p^3$. Building on results of Blackburn and Hall, it is shown that $\lcs G3$ is a maximal subgroup of $G'$, the group $G$ has a central decomposition into two simpler subgroups, and, moreover, $G'$ has one of two isomorphism types.