Finite Group Databasebeta

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Cayley diagram

C2×#10409 #239587

order512
nilpotency class3
isoclinismisoclinic class of #10046
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#10409

Subgroups

subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2541C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

2034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#241691
maximal normal subgroup ofC2×#241691
maximal quotient of#72226863

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146783);