Finite Group Databasebeta

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Cayley diagram

C2×#9109 #196314

order512
nilpotency class3
isoclinismisoclinic class of #9109
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#9109

Subgroups

subgroupG/H
center, socleC23#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#120C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1904 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);(3,4,7,9,6,5,8,10)(11,12)(13,16,20,24)(14,21,22,17)(15,18)(19,23)(25,26);(3,5,7,10,6,4,8,9)(11,13)(12,16,18,24)(14,19)(15,20)(17,23,21,26)(22,25);(4,5)(9,10)(11,14)(12,17)(13,19)(15,22)(16,23)(18,21)(20,25)(24,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240394, C2×#196338, C2×#245567
maximal normal subgroup of#72240394, C2×#196338, C2×#245567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103510);