Finite Group Databasebeta

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Cayley diagram

C2×#11370 #358572

order512
nilpotency class3
isoclinismisoclinic class of #1999
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11370

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C2×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238885, C2×#375492
maximal normal subgroup of#72238885, C2×#375492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265768);