Finite Group Databasebeta

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Cayley diagram

C2×#8092 #181721

order512
nilpotency class3
isoclinismisoclinic class of #327
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#8092

Subgroups

subgroupG/H
centerC2×C42C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1944 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);(3,4,7,10,8,5,6,9)(11,12)(13,21)(14,18)(15,19)(16,20)(17,22)(23,26)(24,25);(3,5,8,4)(6,10,7,9)(11,13,15,23,16,24,14,22)(12,17,18,25,20,26,19,21);(3,6,8,7)(4,9,5,10)(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#181727, #72239955, #72239960, …
maximal normal subgroup ofC2×#181727, #72239955, #72239960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88917);