Groups of order 32

This article gives information about, and links to more details on, groups of order 32
See pages on algebraic structures of order 32 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 32. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 32
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 32
subgroup structure	subgroup structure of groups of order 32
linear representation theory	linear representation theory of groups of order 32 projective representation theory of groups of order 32 modular representation theory of groups of order 32
endomorphism structure, automorphism structure	endomorphism structure of groups of order 32
group cohomology	group cohomology of groups of order 32

Statistics at a glance

Numbers of groups

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

Since $32=2^{5}$ 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	51
Number of abelian groups up to isomorphism	7	Equals the number of unordered integer partitions of 5. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of class exactly two up to isomorphism	26
Number of groups of class exactly three up to isomorphism	15
Number of groups of class exactly four up to isomorphism, 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 32, the groups are: dihedral group:D32, semidihedral group:SD32, and generalized quaternion group:Q32.

Numbers of equivalence classes of groups

Equivalence relation on groups	Number of equivalence classes of groups of order 32	Sizes of equivalence classes, i.e., number of isomorphism classes of groups within each equivalence class (should add up to 51)	More information
isoclinic groups (i.e., Hall-Senior families)	8	7, 15, 10, 9, 2, 2, 3, 3	#Up to isoclinism, see also classification of groups of order 32
isologic groups with respect to nilpotency class two	3	33, 15, 3	#Up to isologism for class two. This is a coarser equivalence relation than being isoclinic.
isologic groups with respect to nilpotency class three	2	48, 3	#Up to isologism for class three. This is a coarser equivalence relation than being isologic with respect to nilpotency class two.
having the same conjugacy class size statistics	6	7,15,19,2,5,3	Element structure of groups of order 32#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	6	7,15,19,2,5,3	See Linear representation theory of groups of order 32#Degrees of irreducible representations. Note that for order 32, the degrees of irreducible representations and the conjugacy class size statistics determine each other, but this breaks down for higher orders. Also, note that this is a coarser equivalence relation than being isoclinic.
1-isomorphic groups	38	1 (29 times), 2 (6 times), 3 (2 times), 4 (1 time)	Element structure of groups of order 32#1-isomorphism

The list

Group	Second part of GAP ID (GAP ID is (32,second part))	Hall-Senior number (among groups of order 32)	Hall-Senior symbol	Nilpotency class	Probability in cohomology tree probability distribution (proper fraction)	Probability in cohomology tree probability distribution (as numerical value)
Cyclic group:Z32	1	7	$(5)$	1	1/16	0.0625
SmallGroup(32,2)	2	18	$\Gamma _{2}h$	2	59/2048	0.0288
Direct product of Z8 and Z4	3	5	$(32)$	1	51/1024	0.0498
Semidirect product of Z8 and Z4 of M-type	4	19	$\Gamma _{2}i$	2	49/1024	0.0479
SmallGroup(32,5)	5	20	$\Gamma _{2}j_{1}$	2	71/1024	0.0693
Faithful semidirect product of E8 and Z4	6	46	$\Gamma _{7}a_{1}$	3	13/1024	0.0127
SmallGroup(32,7)	7	47	$\Gamma _{7}a_{2}$	3	13/2048	0.0063
SmallGroup(32,8)	8	48	$\Gamma _{7}a_{3}$	3	13/2048	0.0063
SmallGroup(32,9)	9	27	$\Gamma _{3}c_{1}$	3	31/1024	0.0303
SmallGroup(32,10)	10	28	$\Gamma _{3}c_{2}$	3	37/1024	0.0361
Wreath product of Z4 and Z2	11	31	$\Gamma _{3}e$	3	13/512	0.0254
SmallGroup(32,12)	12	21	$\Gamma _{2}j_{2}$	2	45/512	0.0879
Semidirect product of Z8 and Z4 of semidihedral type	13	30	$\Gamma _{3}d_{2}$	3	7/256	0.0273
Semidirect product of Z8 and Z4 of dihedral type	14	29	$\Gamma _{3}d_{1}$	3	25/1024	0.0244
SmallGroup(32,15)	15	32	$\Gamma _{3}f$	3	1/32	0.0313
Direct product of Z16 and Z2	16	6	$(41)$	1	31/256	0.1211
M32	17	22	$\Gamma _{2}k$	2	15/256	0.0586
Dihedral group:D32	18	49	$\Gamma _{8}a_{1}$	4	3/1024	0.0029
Semidihedral group:SD32	19	50	$\Gamma _{8}a_{2}$	4	3/512	0.0059
Generalized quaternion group:Q32	20	51	$\Gamma _{8}a_{3}$	4	3/1024	0.0029
Direct product of Z4 and Z4 and Z2	21	3	$(2^{2}1)$	1	637/65536	0.0097
Direct product of SmallGroup(16,3) and Z2	22	11	$\Gamma _{2}c_{1}$	2	695/65536	0.0106
Direct product of SmallGroup(16,4) and Z2	23	12	$\Gamma _{2}c_{2}$	2	349/16384	0.0213
SmallGroup(32,24)	24	16	$\Gamma _{2}f$	2	273/32768	0.0083
Direct product of D8 and Z4	25	14	$\Gamma _{2}e_{1}$	2	69/4096	0.0168
Direct product of Q8 and Z4	26	15	$\Gamma _{2}e_{2}$	2	123/16384	0.0075
SmallGroup(32,27)	27	33	$\Gamma _{4}a_{1}$	2	45/16384	0.0027
SmallGroup(32,28)	28	36	$\Gamma _{4}b_{1}$	2	33/4096	0.0081
SmallGroup(32,29)	29	37	$\Gamma _{4}b_{2}$	2	225/16384	0.0137
SmallGroup(32,30)	30	38	$\Gamma _{4}c_{1}$	2	129/16384	0.0079
SmallGroup(32,31)	31	39	$\Gamma _{4}c_{2}$	2	129/32768	0.0039
SmallGroup(32,32)	32	40	$\Gamma _{4}c_{3}$	2	111/16384	0.0068
SmallGroup(32,33)	33	41	$\Gamma _{4}d$	2	21/4096	0.0051
Generalized dihedral group for direct product of Z4 and Z4	34	34	$\Gamma _{4}a_{2}$	2	45/65536	0.0007
SmallGroup(32,35)	35	35	$\Gamma _{4}a_{3}$	2	321/65536	0.0049
Direct product of Z8 and V4	36	4	$(31^{2})$	1	543/16384	0.0331
Direct product of M16 and Z2	37	13	$\Gamma _{2}d$	2	637/16384	0.0389
Central product of D8 and Z8	38	17	$\Gamma _{2}g$	2	63/4096	0.0154
Direct product of D16 and Z2	39	23	$\Gamma _{3}a_{1}$	3	141/32768	0.0043
Direct product of SD16 and Z2	40	24	$\Gamma _{3}a_{2}$	3	237/16384	0.0145
Direct product of Q16 and Z2	41	25	$\Gamma _{3}a_{3}$	3	237/32768	0.0072
Central product of D16 and Z4	42	26	$\Gamma _{3}b$	3	45/8192	0.0055
Holomorph of Z8	43	44	$\Gamma _{6}a_{1}$	3	45/8192	0.0055
SmallGroup(32,44)	44	45	$\Gamma _{6}a_{2}$	3	45/8192	0.0055
Direct product of E8 and Z4	45	2	$(21^{3})$	1	1023/1048576	0.0010
Direct product of D8 and V4	46	8	$\Gamma _{2}a_{1}$	2	825/1048576	0.0008
Direct product of Q8 and V4	47	9	$\Gamma _{2}a_{2}$	2	771/1048576	0.0007
Direct product of SmallGroup(16,13) and Z2	48	10	$\Gamma _{2}b$	2	329/262144	0.0013
Inner holomorph of D8	49	42	$\Gamma _{5}a_{1}$	2	35/131072	0.0003
Central product of D8 and Q8	50	43	$\Gamma _{5}a_{2}$	2	21/131072	0.0002
Elementary abelian group:E32	51	1	$(1^{5})$	1	1/1048576	0.0000

Arithmetic functions

Summary information

Here, the rows are arithmetic functions that take values between $0$ and $5$, 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 $51$. To view a list of all groups, click on the value in the cell and the list of all groups with GAP IDs appears.

Arithmetic function	Value 0	Value 1	Value 2	Value 3	Value 4	Value 5	Mean (with equal weighting on all groups)	Mean (with weighting by cohomology tree probability distribution)
prime-base logarithm of exponent	0	1	23	21	5	1	2.6471	3.1426
Frattini length	0	1	23	21	5	1	2.6471	3.1426
nilpotency class	0	7	26	15	3	0	2.2745	1.9889
derived length	0	7	44	0	0	0	1.8627	1.7228
minimum size of generating set (sometimes called rank, though it differs from rank of a p-group as used below)	0	1	19	24	6	1	2.7451	2.2039
rank of a p-group	0	2	21	23	4	1	2.6275	2.3064
normal rank of a p-group	0	4	23	19	4	1	2.5098	2.2431
characteristic rank of a p-group	0	7	26	14	3	1	2.3137	2.1972

Families and classification

Up to isoclinism

FACTS TO CHECK AGAINST for isoclinic groups (groups with an isoclinism between them):
by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.
isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations

The information below collects groups based on the equivalence relation of being isoclinic groups. The equivalence classes are also called Hall-Senior families.

Family name	Isomorphism class of inner automorphism group	Isomorphism class of derived subgroup	Number of members	Nilpotency class	Members	Second part of GAP ID of members (sorted ascending)	Hall-Senior numbers of members (sorted ascending)
$\Gamma _{1}$	trivial group	trivial group	7	1	all abelian groups of order 32: [SHOW MORE] cyclic group:Z32, direct product of Z8 and Z4, direct product of Z16 and Z2, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of Z4 and E8, elementary abelian group:E32	1,3,16,21,36,45,51	1-7
$\Gamma _{2}$	Klein four-group	cyclic group:Z2	15	2	[SHOW MORE] direct product of D8 and V4, direct product of Q8 and V4, direct product of SmallGroup(16,13) and Z2, direct product of SmallGroup(16,3) and Z2, direct product of SmallGroup(16,4) and Z2, direct product of M16 and Z2, direct product of D8 and Z4, direct product of Q8 and Z4, SmallGroup(32,24), central product of D8 and Z8, SmallGroup(32,2), SmallGroup(32,5), SmallGroup(32,12), SmallGroup(32,12), M32, semidirect product of Z8 and Z4 of M-type	2,4,5,12,17,22,23,24,25,26,37,38,46,47,48	8-22
$\Gamma _{3}$	dihedral group:D8	cyclic group:Z4	10	3	[SHOW MORE] direct product of D16 and Z2, direct product of SD16 and Z2, direct product of Q16 and Z2, central product of D16 and Z4, semidirect product of Z8 and Z4 of dihedral type, semidirect product of Z8 and Z4 of semidihedral type, SmallGroup(32,9), SmallGroup(32,10), wreath product of Z4 and Z2, SmallGroup(32,15)	9,10,11,13,14,15,39,40,41,42	23-32
$\Gamma _{4}$	elementary abelian group:E8	Klein four-group	9	2	[SHOW MORE] SmallGroup(32,27), SmallGroup(32,28), SmallGroup(32,29), SmallGroup(32,30), SmallGroup(32,31), SmallGroup(32,32), SmallGroup(32,33), generalized dihedral group for direct product of Z4 and Z4, SmallGroup(32,35)	27-35	33-41
$\Gamma _{5}$	elementary abelian group:E16	cyclic group:Z2	2	2	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43
$\Gamma _{6}$	direct product of D8 and Z2	cyclic group:Z4	2	3	holomorph of Z8, SmallGroup(32,44)	43,44	44,45
$\Gamma _{7}$	SmallGroup(16,3)	Klein four-group	3	3	[SHOW MORE] faithful semidirect product of E8 and Z4, SmallGroup(32,7), SmallGroup(32,8)	6-8	46-48
$\Gamma _{8}$	dihedral group:D16	cyclic group:Z8	3	4	[SHOW MORE] dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18-20	49-51
Total (8 rows)	--	--	51	--	--	--	--

Up to Hall-Senior genus

Genus name	Members	Second part of GAP ID of members	Hall-Senior numbers of members
$(1^{5})$	elementary abelian group:E32	51	1
$(21^{3})$	direct product of E8 and Z4	45	2
$(2^{2}1)$	direct product of Z4 and Z4 and Z2	21	3
$(31^{2})$	direct product of Z8 and V4	36	4
$(32)$	direct product of Z8 and Z4	3	5
$(41)$	direct product of Z16 and Z2	16	6
$(5)$	cyclic group:Z32	1	7
$\Gamma _{2}a$	direct product of D8 and V4, direct product of Q8 and V4	46,47	8,9
$\Gamma _{2}b$	direct product of SmallGroup(16,13) and Z2	48	10
$\Gamma _{2}c$	direct product of SmallGroup(16,3) and Z2, direct product of SmallGroup(16,4) and Z2	22,23	11,12
$\Gamma _{2}d$	direct product of M16 and Z2	37	13
$\Gamma _{2}e$	direct product of D8 and Z4, direct product of Q8 and Z4	25, 26	14, 15
$\Gamma _{2}f$	SmallGroup(32,24)	24	16
$\Gamma _{2}g$	central product of D8 and Z8	38	17
$\Gamma _{2}h$	SmallGroup(32,2)	2	18
$\Gamma _{2}i$	semidirect product of Z8 and Z4 of M-type	4	19
$\Gamma _{2}j$	SmallGroup(32,5), SmallGroup(32,12)	5, 12	20, 21
$\Gamma _{2}k$	M32	17	22
$\Gamma _{3}a$	direct product of D16 and Z2, direct product of SD16 and Z2, direct product of Q16 and Z2	39,40,41	23,24,25
$\Gamma _{3}b$	central product of D8 and Z8	42	26
$\Gamma _{3}c$	SmallGroup(32,9), SmallGroup(32,10)	9,10	27,28
$\Gamma _{3}d$	semidirect product of Z8 and Z4 of semidihedral type, semidirect product of Z8 and Z4 of dihedral type	13,14	29,30
$\Gamma _{3}e$	wreath product of Z4 and Z2	11	31
$\Gamma _{3}f$	SmallGroup(32,15)	15	32
$\Gamma _{4}a$	SmallGroup(32,27), generalized dihedral group for direct product of Z4 and Z4, SmallGroup(32,35)	27, 34, 35	33, 34, 35
$\Gamma _{4}b$	SmallGroup(32,28), SmallGroup(32,29)	28,29	36,37
$\Gamma _{4}c$	SmallGroup(32,30), SmallGroup(32,31), SmallGroup(32,32)	30,31,32	38,39,40
$\Gamma _{4}d$	SmallGroup(32,33)	33	41
$\Gamma _{5}a$	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43
$\Gamma _{6}a$	holomorph of Z8, SmallGroup(32,44)	43, 44	44, 45
$\Gamma _{7}a$	faithful semidirect product of E8 and Z4, SmallGroup(32,7), SmallGroup(32,8)	6, 7, 8	46, 47, 48
$\Gamma _{8}a$	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18, 19, 20	49, 50 51

Up to isologism for class two

Under the equivalence relation of being isologic groups with respect to the variety of groups of nilpotency class two, the equivalence classes are as follows (the table is incomplete):

Isomorphism class of quotient by second center	Isomorphism class of third member of lower central series	Number of groups	Nilpotency class(es)	Second part of GAP ID of members (sorted ascending)	Hall-Senior numbers of members (sorted ascending)	Smallest order of group isologic to these groups	Stem groups (groups of smallest order)
trivial group	trivial group	33	1,2	1-5,12,16,17,21-38,45-51	1-22,33-43	1	trivial group
Klein four-group	cyclic group:Z2	15	3	6-11, 13-15, 39-44	23-32, 44-48	16	dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16
dihedral group:D8	cyclic group:Z4	3	4	18-20	49-51	32	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32
-- (3 rows)	--	51	--	--	--	--	--

Up to isologism for class three

Under the equivalence relation of being isologic groups with respect to the variety of groups of class at most three, there are two equivalence classes:

Isomorphism class of quotient by third center	Isomorphism class of fourth member of lower central series	Number of groups	Nilpotency class(es)	Second part of GAP ID of members (sorted ascending)	Hall-Senior numbers of members (sorted ascending)	Smallest order of group isologic to these groups	Stem groups (groups of smallest order)
trivial group	trivial group	48	1,2,3	1-17,21-51	1-48	1	trivial group
Klein four-group	cyclic group:Z2	3	4	18-20	49-51	32	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32
-- (2 rows)	--	51	--	--	--	--	--

Up to isologism for higher class

For class four or higher, all groups of order 32 are isologic to each other.

Element structure

Further information: element structure of groups of order 32

Subgroup structure

Further information: subgroup structure of groups of order 32

Linear representation theory

Further information: linear representation theory of groups of order 32

References

• First complete published classification: The regular substitution groups whose orders are less than 48 by G. A. Miller, Quarterly Journal of Mathematics, Volume 28, Page 232 - 284(Year 1896): ^{More info}
• Detailed information about the groups: 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 32 is part of GAP's SmallGroup library. Hence, any group of order 32 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 32 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(32);

  There are 51 groups of order 32.
  They are sorted by their ranks.
     1 is cyclic.
     2 - 20 have rank 2.
     21 - 44 have rank 3.
     45 - 50 have rank 4.
     51 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.