Finite Group Databasebeta

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Cayley diagram

C2×#8807 #191369

order512
nilpotency class3
isoclinismisoclinic class of #465
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#8807

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socle, prime radicalC24C2×C4⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2);(4,6)(7,8,9,11,10,12,13,14)(15,16)(17,23)(18,26)(19,21)(20,24)(22,27)(25,29)(28,30);(15,17,18,24,19,25,27,30)(16,20,22,23,21,28,26,29);(3,4,5,6)(16,21)(20,28)(22,26)(23,29)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210318, #72210445, C2×#135716
maximal normal subgroup of#72210318, #72210445, C2×#135716
maximal quotient ofC2×#135732

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98565);