Finite Group Databasebeta

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Cayley diagram

C2×#23877 #6437467

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C211

Aliases

direct productC2×#23877

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3488C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

6348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72317709, C2×#6499844, #72318128
maximal normal subgroup of#72317709, C2×#6499844, #72318128

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344663);