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Finding non-nilpotent groups of odd order without non-cyclic nilpotent quotients

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?
show/hide this revision's text 1

Finding non-nilpotent groups of odd order

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 $\leq 2015$.