Finite Group Databasebeta

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Cayley diagram

C2×#9523 #201988

order512
nilpotency class3
isoclinismisoclinic class of #9519
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9523

Subgroups

subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3006C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2254 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)(5,9,10,7)(6,8)(11,12,15,19,16,20,14,17)(13,21,23,18,24,25,22,26);(3,4)(5,7)(6,8)(9,10)(12,17,20,19)(18,21,26,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,22)(15,23)(16,24)(17,21)(19,25)(20,26)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#202007, C2×#201997, C2×#202016
maximal normal subgroup ofC2×#202007, C2×#201997, C2×#202016
maximal quotient of#72309101

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109184);