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

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
Number of groups up to isomorphism	51
Number of abelian groups up to isomorphism	7
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	3

The list

Note there's an ambiguity that makes the table below incomplete: the Hall-Senior numbers of groups with GAP IDs 13 and 14 are 29 and 30 (symbol ${\mathrm{\Gamma }}_3{d}_1$ and ${\mathrm{\Gamma }}_3{d}_2$ respectively) but it's not yet clear which GAP ID corresponds to which Hall-Senior number.

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
Cyclic group:Z32	1	7	$(5)$	1
SmallGroup(32,2)	2	18	${\mathrm{\Gamma }}_{2}h$	2
Direct product of Z8 and Z4	3	5	$(32)$	1
Semidirect product of Z8 and Z4 of M-type	4	19	${\mathrm{\Gamma }}_{2}i$	2
SmallGroup(32,5)	5	20	${\mathrm{\Gamma }}_2{j}_1$	2
Faithful semidirect product of E8 and Z4	6	46	${\mathrm{\Gamma }}_7{a}_1$	3
SmallGroup(32,7)	7	47	${\mathrm{\Gamma }}_7{a}_2$	3
SmallGroup(32,8)	8	48	${\mathrm{\Gamma }}_7{a}_3$	3
SmallGroup(32,9)	9	27	${\mathrm{\Gamma }}_3{c}_1$	3
SmallGroup(32,10)	10	28	${\mathrm{\Gamma }}_3{c}_2$	3
Wreath product of Z4 and Z2	11	31	${\mathrm{\Gamma }}_{3}e$	3
SmallGroup(32,12)	12	21	${\mathrm{\Gamma }}_2{j}_2$	2
Semidirect product of Z8 and Z4 of semidihedral type	13			3
Semidirect product of Z8 and Z4 of dihedral type	14			3
SmallGroup(32,15)	15	32	${\mathrm{\Gamma }}_{3}f$	3
Direct product of Z16 and Z2	16	6	$(41)$	1
M32	17	22	${\mathrm{\Gamma }}_{2}k$	2
Dihedral group:D32	18	49	${\mathrm{\Gamma }}_8{a}_1$	4
Semidihedral group:SD32	19	50	${\mathrm{\Gamma }}_8{a}_2$	4
Generalized quaternion group:Q32	20	51	${\mathrm{\Gamma }}_8{a}_3$	4
Direct product of Z4 and Z4 and Z2	21	3	$(2^{2}1)$	1
Direct product of SmallGroup(16,3) and Z2	22	11	${\mathrm{\Gamma }}_2{c}_1$	2
Direct product of SmallGroup(16,4) and Z2	23	12	${\mathrm{\Gamma }}_2{c}_2$	2
SmallGroup(32,24)	24	16	${\mathrm{\Gamma }}_{2}f$	2
Direct product of D8 and Z4	25	14	${\mathrm{\Gamma }}_2{e}_1$	2
Direct product of Q8 and Z4	26	15	${\mathrm{\Gamma }}_2{e}_2$	2
SmallGroup(32,27)	27	33	${\mathrm{\Gamma }}_4{a}_1$	2
SmallGroup(32,28)	28	36	${\mathrm{\Gamma }}_4{b}_1$	2
SmallGroup(32,29)	29	37	${\mathrm{\Gamma }}_4{b}_2$	2
SmallGroup(32,30)	30	38	${\mathrm{\Gamma }}_4{c}_1$	2
SmallGroup(32,31)	31	39	${\mathrm{\Gamma }}_4{c}_2$	2
SmallGroup(32,32)	32	40	${\mathrm{\Gamma }}_4{c}_3$	2
SmallGroup(32,33)	33	41	${\mathrm{\Gamma }}_{4}d$	2
Generalized dihedral group for direct product of Z4 and Z4	34	34	${\mathrm{\Gamma }}_4{a}_2$	2
SmallGroup(32,35)	35	35	${\mathrm{\Gamma }}_4{a}_3$	2
Direct product of Z8 and V4	36	4	$(31^2)$	1
Direct product of M16 and Z2	37	13	${\mathrm{\Gamma }}_{2}d$	2
Central product of D8 and Z8	38	17	${\mathrm{\Gamma }}_{2}g$	2
Direct product of D16 and Z2	39	23	${\mathrm{\Gamma }}_3{a}_1$	3
Direct product of SD16 and Z2	40	24	${\mathrm{\Gamma }}_3{a}_2$	3
Direct product of Q16 and Z2	41	25	${\mathrm{\Gamma }}_3{a}_3$	3
Central product of D16 and Z4	42	26	${\mathrm{\Gamma }}_{3}b$	3
Holomorph of Z8	43	44	${\mathrm{\Gamma }}_6{a}_1$	3
SmallGroup(32,44)	44	45	${\mathrm{\Gamma }}_6{a}_2$	3
Direct product of E8 and Z4	45	2	$(21^3)$	1
Direct product of D8 and V4	46	8	${\mathrm{\Gamma }}_2{a}_1$	2
Direct product of Q8 and V4	47	9	${\mathrm{\Gamma }}_2{a}_2$	2
Direct product of SmallGroup(16,13) and Z2	48	10	${\mathrm{\Gamma }}_{2}b$	2
Inner holomorph of D8	49	42	${\mathrm{\Gamma }}_5{a}_1$	2
Central product of D8 and Q8	50	43	${\mathrm{\Gamma }}_5{a}_2$	2
Elementary abelian group:E32	51	1	$(1^5)$	1

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
prime-base logarithm of exponent	0	1	23	21	5	1
Frattini length	0	1	23	21	5	1
nilpotency class	0	7	26	15	3	0
derived length	0	7	44	0	0	0
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
rank of a p-group	0	2	21	23	4	1
normal rank of a p-group	0	4	23	19	4	1
characteristic rank of a p-group	0	7	26	14	3	1

Families and classification

Isocliny, or 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)
${\mathrm{\Gamma }}_1$	trivial group	trivial group	7	1	all abelian groups of order 32: 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
${\mathrm{\Gamma }}_2$	Klein four-group	cyclic group:Z2	15	2	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
${\mathrm{\Gamma }}_3$	dihedral group:D8	cyclic group:Z4	10	3	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
${\mathrm{\Gamma }}_4$	elementary abelian group:E8	Klein four-group	9	2	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
${\mathrm{\Gamma }}_5$	elementary abelian group:E16	cyclic group:Z2	2	2	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43
${\mathrm{\Gamma }}_6$	direct product of D8 and Z2	cyclic group:Z2	2	3	holomorph of Z8, SmallGroup(32,44)	43,44	44,45
${\mathrm{\Gamma }}_7$	SmallGroup(16,3)	Klein four-group	3	3	faithful semidirect product of E8 and Z4, SmallGroup(32,7), SmallGroup(32,8)	6-8	46-48
${\mathrm{\Gamma }}_8$	dihedral group:D16	cyclic group:Z8	3	4	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18-20	49-51

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

Element structure

Further information: element structure of groups of order 32