Finite Group Databasebeta

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Cayley diagram

C2×#11125 #357804

order512
nilpotency class3
isoclinismisoclinic class of #1999
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#11125

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C4×D4
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72215286, #72215970, C2×#378848
maximal normal subgroup of#72215286, #72215970, C2×#378848
maximal quotient ofC2×#117930, #72215279

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265000);