Characteristic rank of a p-group

Definition
The characteristic rank of a p-group is defined in the following equivalent ways:


 * It is the largest $$r$$ for which there exists an elementary Abelian characteristic subgroup of order $$p^r$$
 * It is the largest $$r$$ for which there exists an Abelian characteristic subgroup for which the minimum size of a generating set is $$r$$

Related notions

 * Rank of a p-group
 * Normal rank of a p-group