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?
Classification of finite nilpotent groups of maximal class
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0What is that "maximal nilpotency class"?? – 2012-06-27
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1@DonAntonio: A group of order $p^n$ will have class at most $n-1$; a $p$-group is said to be "of maximal class" if its order is $p^n$ and the class is *exactly* $n-1$. The $p$-groups of maximal class were essentially described by Blackburn, and the Coclass Conjecture programme (now theorems) was based on attempting to emulate his ideas for more general $p$-groups. – 2012-06-27
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0@Arturo: Provided $n\ge 2$. – 2012-06-27
2 Answers
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.