Classification of finite p-groups of characteristic rank one

Definition
Suppose $$G$$ is a group of prime power order that is also of characteristic rank one. Then, $$G$$ is expressible asa central product $$ER$$, such that:


 * $$E$$ is either trivial or extraspecial
 * $$R$$ is a cyclic group if $$p$$ is an odd prime. For $$p=2$$, $$R$$ is either cyclic, or has a cyclic maximal subgroup with the quotient acting on it by multiplication by either $$-1$$ or $$2^{r-2} - 1$$, where $$|R| = 2^r$$. In particular, it has a cyclic maximal subgroup but is not of the form where the quotient acts via multiplication by $$2^{r-2} + 1$$.

Related facts

 * Classification of finite p-groups of normal rank one
 * Classification of finite p-groups of rank one

Textbook references

 * , Page 198-199, Theorem 4.9, Section 5.4 ($$p$$-groups of small depth)