Exponential rank of a finite p-group

Definition
Suppose $$p$$ is a prime number and $$P$$ is a finite p-group. The exponential rank of $$P$$ is defined as follows.

We know that the exponent semigroup $$\mathcal{E}(P)$$ is characterized in terms of a nonnegative integer $$m$$ as all multiples of $$p^m$$ union with all numbers that are 1 more than multiples of $$p^m$$. This $$m$$ is uniquely determined and is at least equal to $$\log_p(\mbox{exponent of } P/Z(P))$$.

The exponential rank of $$P$$ is the number:

$$m - \log_p(\mbox{exponent of } P/Z(P))$$

Facts

 * Finite abelian p-groups have exponential rank zero.
 * Regular p-groups (and hence, finite $$p$$-groups of nilpotency class less than p, have exponential rank zero.
 * For $$p = 2$$, dihedral group:D8 and quaternion group, both of order eight, both have exponential rank one.