I have seen the page demonstrating that it is practically impossible to classify all nilpotent groups, but could you classify all groups of maximal nilpotency class?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
The coclass of a $p$-group of order $p^n$ is defined to be $n-c$, where $c$ is its nilpotency class. So a $p$-group of maximal class has coclass 1.
Following on from the work of Blackburn on groups of maximal class, there has been a lot of research on $p$-groups and pro-$p$-groups with bounded coclass. The basic source for this topic is the book:
Leedham-Green, C. R.; McKay, Susan (2002), The structure of groups of prime power order, London Mathematical Society Monographs. New Series, 27, Oxford University Press,
Good question, but at the same time a very difficult question to answer. Norman Blackburn started investigations back in 1958. See also the books of B. Huppert Endliche Gruppen (a.k.a Finite Groups I), and B. Huppert & N. Blackburn Finite groups II, both Springer-Verlag, Berlin in which a lot of results can be found.