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Counting p-groups






Here we provide access to functions which count 
the number of d-generator p-class 2 groups.
For details of the algorithm used see [EO99].


SetVerbose ("ClassTwo", 1) will provide
information on the progress of the algorithm.



ClassTwo(p, d : parameters) : RngIntElt, RngIntElt -> SeqEnum


ClassTwo(p, d, Step : parameters) : RngIntElt, RngIntElt, SeqEnum -> SeqEnum


ClassTwo(p, d, s : parameters) : RngIntElt, RngIntElt, RngIntElt -> RngIntElt



Count the d-generator p-groups of p-class 2. 
If s or Step is supplied, then count only those of 
order p^((d + s)) or p^((d + m)) for m in Step.
In the first two invocations, the sequence returns
a sequence of length d choose 2, whose m-th 
entry is the number of groups of p^((d + m)). 
(Some additional entries may be deduced on the basis
of duality.) The last invocation returns the number 
of groups of p^((d + s)).





     Exponent: RngIntElt                 Default: 0




If Exponent is true, count those groups which have exponent p.







Example GrpPGp_ClassTwo (H20E6)

Count the number of 3-generator p-class 2 5-groups.





> ClassTwo(5, 3);
[ 4, 19, 42, 19, 4, 1 ]


For example, the number of 3-generator 5-groups of order 5^6 
and p-class 2 is precisely 42.


Count the number of 4-generator p-class 2 5-groups of order 5^7.





> ClassTwo(5, 4, 3);
6598






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