Finite Group Databasebeta

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Cayley diagram

C2×#11711 #481471

order512
nilpotency class4
isoclinismisoclinic class of #11704
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#11711

Subgroups

subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#466C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#482090, C2×#128808, C2×#128836
maximal normal subgroup ofC2×#482090, C2×#128808, C2×#128836

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388667);