Finite Group Databasebeta

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Cayley diagram

C2×#13316 #498818

order512
nilpotency class4
isoclinismisoclinic class of #13316
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#13316

Subgroups

subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499263, #72224844, #72224849, …
maximal normal subgroup ofC2×#499263, #72224844, #72224849, …
maximal quotient ofC2×#147939, #72224844

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406014);