Finite Group Databasebeta

← Prev (#126767) Next (#126769) →

Cayley diagram

C2×#7171 #126768

order512
nilpotency class4
isoclinismisoclinic class of D16
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25

Aliases

direct productC2×#7171

Subgroups

subgroupG/H
centerC23×C4D8
commutatorC8C2×C4×C8
Frattini, cosocleC2×C4×C8C23
socle, prime radicalC24#102
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2);(3,4)(5,6)(7,8,10,13,11,14,9,12)(16,21)(17,25)(18,27)(19,26)(22,23)(24,30)(28,29);(4,6)(8,12,14,13)(15,16,18,22,19,23,17,21,20,24,25,29,26,28,27,30)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210445, C2×#128531, C2×C8×#357, …
maximal normal subgroup of#72210445, C2×#128531, C2×C8×#357, …
maximal quotient of#72210437, C2×#94047, C2×#94048, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33964);