Finite Group Databasebeta

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Cayley diagram

C2×#9940 #234924

order512
nilpotency class3
isoclinismisoclinic class of #9938
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#9940

Subgroups

subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2539C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#234973
maximal normal subgroup ofC2×#234973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142120);