Groups of order 256

This article gives information about, and links to more details on, groups of order 256
See pages on algebraic structures of order 256 | See pages on groups of a particular order

Statistics at a glance

Note that since prime power order implies nilpotent, and $256=2^{8}$ is a prime power, all groups of order 256 are nilpotent.

Quantity	Value	Greatest integer function of logarithm of value to base 2	Explanation for value
Number of groups up to isomorphism	56092	15
Number of abelian groups up to isomorphism	22	4	Equal to the number of unordered integer partitions of 8, see classification of finite abelian groups
Number of groups of class exactly two up to isomorphism	31742	14
Number of groups of class exactly three up to isomorphism	21325	14
Number of groups of class exactly four up to isomorphism	2642	11
Number of groups of class exactly five up to isomorphism	320	8
Number of groups of class exactly six up to isomorphism	38	5
Number of groups of class exactly seven up to isomorphism, i.e., maximal class groups	3	1	classification of finite 2-groups of maximal class. For order $2^{n},n\geq 4$ , there are exactly three maximal class groups: dihedral, semidihedral, and generalized quaternion. For order 256, the groups are: dihedral group:D256, semidihedral group:SD256, and generalized quaternion group:Q256.

References

The groups of order 256 by E. A. O'Brien, Journal of Algebra, ISSN 00218693, Volume 143, Page 219 - 235(Year 1991): ^{Official copy (Elsevier ScienceDirect)}^{More info}

GAP implementation

The order 256 is part of GAP's SmallGroup library. Hence, any group of order 256 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 256 can be accessed as a list using GAP's AllSmallGroups function. However, the list size may be too large relative to the memory allocation given in typical GAP installations. To overcome this problem, use the IdsOfAllSmallGroups function which stores and manipulates only the group IDs, not the groups themselves.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(256);

  There are 56092 groups of order 256.
  They are sorted by their ranks.
     1 is cyclic.
     2 - 541 have rank 2.
     542 - 6731 have rank 3.
     6732 - 26972 have rank 4.
     26973 - 55625 have rank 5.
     55626 - 56081 have rank 6.
     56082 - 56091 have rank 7.
     56092 is elementary abelian.

  For the selection functions the values of the following attributes
  are precomputed and stored:
     IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
     FrattinifactorId.

  This size belongs to layer 2 of the SmallGroups library.
  IdSmallGroup is available for this size.