Finite Group Databasebeta

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Cayley diagram

C2×#10893 #353911

order512
nilpotency class3
isoclinismisoclinic class of #453
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#10893

Subgroups

subgroupG/H
centerC22×C4#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23#386
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2190 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);(3,4,6,5)(7,8)(9,16)(10,19)(11,14)(12,15)(13,17)(18,22)(20,21);(7,9,11,17,12,18,10,20)(8,13,14,22,15,21,19,16);(3,5,6,4)(7,8)(9,13)(10,19)(11,14)(12,15)(16,20)(17,22)(18,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235153, #72238844
maximal normal subgroup of#72235153, #72238844
maximal quotient ofC2×#123488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261107);