Finite Group Databasebeta

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Cayley diagram

C2×#11895 #482082

order512
nilpotency class4
isoclinismisoclinic class of #11858
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#11895

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

1238 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)(11,12,15,20,17,22,16,21)(13,18,23,19,14,25,24,26);(3,5)(4,7)(6,10)(8,9)(12,18)(13,23)(14,24)(15,16)(19,21)(20,26)(22,25);(5,10)(7,9)(11,13,16,24,17,14,15,23)(12,19,20,25,22,26,21,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#482090, C2×#130722
maximal normal subgroup ofC2×#482090, C2×#130722

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389278);