Finite Group Databasebeta

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Cayley diagram

C2×#11888 #482067

order512
nilpotency class4
isoclinismisoclinic class of #11858
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#11888

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

1294 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)(13,21)(14,17)(15,25)(16,20)(18,23)(19,24)(22,26);(3,5)(4,7)(6,10)(8,9)(12,17)(13,22)(14,23)(15,24)(18,19)(20,26)(21,25);(5,10)(7,9)(11,13,15,14,16,23,24,22)(12,17,19,21,20,26,25,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#482075
maximal normal subgroup ofC2×#482075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389263);