Finite Group Databasebeta

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Cayley diagram

C2×#8612 #188833

order512
nilpotency class3
isoclinismisoclinic class of #8608
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#8612

Subgroups

subgroupG/H
center, socleC23#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1346 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)(11,12,15,18,16,19,14,24)(13,20,22,25,23,26,21,17);(11,13,15,22,16,23,14,21)(12,17,24,26,19,25,18,20);(3,5)(4,7)(6,10)(8,9)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#188834
maximal normal subgroup ofC2×#188834

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96029);