Groups of order 64

This article gives information about, and links to more details on, groups of order 64
See pages on algebraic structures of order 64 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 64. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 64
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 64
subgroup structure	subgroup structure of groups of order 64
linear representation theory	linear representation theory of groups of order 64 projective representation theory of groups of order 64 modular representation theory of groups of order 64
endomorphism structure, automorphism structure	endomorphism structure of groups of order 64
group cohomology	group cohomology of groups of order 64

Statistics at a glance

To understand these in a broader context, see
groups of order 2^n|groups of prime-sixth order

Numbers of groups

Since $64=2^{6}$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity	Value	Explanation
Number of groups up to isomorphism	267
Number of abelian groups up to isomorphism	11	Equals the number of unordered integer partitions of $6$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of class exactly two	117
Number of groups of class exactly three	114
Number of groups of class exactly four	22
Number of groups of class exactly five, i.e., maximal class groups	3	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 64, the groups are: dihedral group:D64, semidihedral group:SD64, and generalized quaternion group:Q64.

Numbers of equivalence classes of groups

Equivalence relation on groups	Number of equivalence classes of groups of order 64	Sizes of equivalence classes, i.e., number of isomorphism classes of groups within each equivalence class (should add up to 267)	More information
isoclinic groups (i.e., Hall-Senior families)	27	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	#Families and classification, see also classification of groups of order 64
having the same conjugacy class size statistics	16	6, 6, 9, 3, 3, 9, 31, 38, 15, 23, 7, 5, 60, 10, 31, 11	Element structure of groups of order 64#Conjugacy class sizes. Note that isoclinic groups of the same order have the same conjugacy class size statistics, so this is a coarser equivalence relation than being isoclinic.
having the same degrees of irreducible representations	12	3, 6, 24, 31, 33, 41, 15, 60, 5, 31, 7, 11	See Linear representation theory of groups of order 64#Degrees of irreducible representations. Note that for order 64, this gives a coarser equivalence relation than that arising from the same conjugacy class sizes, but for higher orders, the equivalence relations from conjugacy class sizes and degrees of irreducible representations are incomparable. Also, note that this is a coarser equivalence relation than being isoclinic (and that is a fact true for all orders, because isoclinic groups of the same order have the same degrees of irreducible representations).
1-isomorphic groups			Element structure of groups of order 64#1-isomorphism

The list

The list is very long (267) for this order, so it has been given its own page on the wiki. See list of groups of order 64.

Arithmetic functions

Summary information

Here, the rows are arithmetic functions that take values between $0$ and $6$ , and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal $267$ , which is the total number of groups of order $64$ .

Arithmetic function	Value 0	Value 1	Value 2	Value 3	Value 4	Value 5	Value 6
prime-base logarithm of exponent	0	1	96	137	27	5	1
Frattini length	0	1	94	139	27	5	1
nilpotency class	0	11	117	114	22	3	0
derived length	0	11	256	0	0	0	0
minimum size of generating set	0	1	53	137	68	7	1
rank of a p-group	0	2	54	150	55	5	1
normal rank of a p-group	0	4	87	122	48	5	1
characteristic rank of a p-group

References

The groups of order $2^{n}$ ( $n\leq 6$ ) by Marshall Hall and James Kuhn Senior. Reviewed on MathScinet^{More info}

GAP implementation

The order 64 is part of GAP's SmallGroup library. Hence, any group of order 64 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 64 can be accessed as a list using GAP's AllSmallGroups function.

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

gap> SmallGroupsInformation(64);

  There are 267 groups of order 64.
  They are sorted by their ranks.
     1 is cyclic.
     2 - 54 have rank 2.
     55 - 191 have rank 3.
     192 - 259 have rank 4.
     260 - 266 have rank 5.
     267 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.</pre.