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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?

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What is that "maximal nilpotency class"?? –  DonAntonio Jun 27 '12 at 12:18
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@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. –  Arturo Magidin Jun 27 '12 at 16:44
    
@Arturo: Provided $n\ge 2$. –  Rod Jun 27 '12 at 17:10
    
Thank you, @ArturoMagidin –  DonAntonio Jun 27 '12 at 21:56
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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,

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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.

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