Finite Group Databasebeta

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Cayley diagram

C2×#8378 #185618

order512
nilpotency class3
isoclinismisoclinic class of #8359
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#8378

Subgroups

subgroupG/H
center, socleC23#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#553C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2726 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);(3,4,7,9,8,5,6,10)(11,12)(13,19,21,16)(14,22,23,17)(15,18)(20,24)(25,26);(3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,20)(15,21)(17,24)(18,19)(22,25)(23,26);(3,6,8,7)(4,9,5,10)(11,14,15,23)(12,17,18,22)(13,20,21,26)(16,24,19,25)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#185657, #72313341, #72313349
maximal normal subgroup ofC2×#185657, #72313341, #72313349
maximal quotient ofC2×#106162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92814);