I hope that my question is appropriate for MO, since it might turn out te be mainly a question about GAP or other group theory software.
Is there an algorithm to produce all non-nilpotent groups of odd order (up to some given upper bound)?
All groups of odd order are solvable by the famous Feit-Thompson theorem; I guess that this fact could be useful in enumerating all these groups, but I don't know how.
Any general ideas to produce many such groups (not necessarily all of them) would also be greatly appreciated.
I already checked the SmallGroups GAP database, and it turns out that there are only 1016 such groups of order $\leq 2015$.
EDIT: The following question probably makes more sense:
Is there an algorithm to produce all groups of odd order (up to some given upper bound) not admitting a non-cyclic nilpotent quotient?