[erfanian@wali.um.ac.ir: [GAP Forum] looking for a group]

[Mike Newman]

It has been pointed out to me that my recent response was flawed.
There are more examples.
All groups with prime-power order with derived subgroup of prime order
are examples.

For a complete determination of these groups see

Blackburn, Simon R. Groups of prime power order with derived subgroup of
prime order. J. Algebra 219 (1999), no. 2, 625--657

      Mike Newman