Groups of order 64

See pages on algebraic structures of order 64| See pages on groups of a particular order

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 $2^n, n \ge 4$, 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] #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

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 $2^n$ ($n \le 6$) by Marshall Hall and James Kuhn Senior. Reviewed on MathScinetMore 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.```