Finite Group Databasebeta

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Cayley diagram

C2×#7982 #181235

order512
nilpotency class3
isoclinismisoclinic class of #1284
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#7982

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#95
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

3194 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)(13,21)(14,18)(15,19)(16,20)(17,22)(23,26)(24,25);(3,5)(4,7)(6,10)(8,9)(11,13,15,23,16,24,14,22)(12,17,18,25,20,26,19,21);(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#188806, C2×#108573, C2×#105326, …
maximal normal subgroup ofC2×#188806, C2×#108573, C2×#105326, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88431);