Extraspecial group

Definition
A group of prime power order (or, more generally, any p-group) is termed extraspecial if its center, derived subgroup and Frattini subgroup all coincide, and moreover, each of these is a group of prime order (and hence, a cyclic group).

Classification and particular cases
Any finite extraspecial group has order $$p^{2r + 1}$$ for some positive integer $$r$$. For each $$r$$, there are two extraspecial groups (up to isomorphism), denoted the + and - types respectively.

The case $$p = 2$$
All the extraspecial groups are obtained as iterated central products of copies of dihedral group:D8 and quaternion group. The $$+$$ type corresponds to the cases where the quaternion group part occurs an even number of times in the central product and the $$-$$ type corresponds to the case where the quaternion group occurs an odd number of times.

Weaker properties

 * Stronger than::Special group
 * Stronger than::Frattini-in-center group
 * Stronger than::Group of nilpotency class two
 * Stronger than::Camina group: