Finite Group Databasebeta

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Cayley diagram

C2×#11085 #357755

order512
nilpotency class3
isoclinismisoclinic class of #2031
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#11085

Subgroups

subgroupG/H
centerC22×C4#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23#393
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(4,6)(7,8)(9,15,17,12)(10,14)(11,13)(16,18,22,19)(20,21);(3,4,5,6)(7,8)(9,16,17,22)(10,14)(11,13)(12,18,15,19)(20,21);(7,9)(8,12)(10,19)(11,17)(13,15)(14,22)(16,21)(18,20)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235146, #72235513, C2×#357756
maximal normal subgroup of#72235146, #72235513, C2×#357756
maximal quotient ofC2×#123745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264951);