Finite Group Databasebeta

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Cayley diagram

C2×#9480 #201454

order512
nilpotency class3
isoclinismisoclinic class of #9471
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9480

Subgroups

subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2190 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)(5,9,10,7)(6,8)(11,12,15,19,16,20,14,18)(13,21,22,25,24,26,23,17);(3,4)(5,7)(6,8)(9,10)(11,13)(12,17,20,25)(14,23)(15,22)(16,24)(18,26,19,21);(3,5)(4,7)(6,10)(8,9)(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,21,25,26)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#201463
maximal normal subgroup ofC2×#201463

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108650);