Finite Group Databasebeta

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Cayley diagram

C2×#21416 #6372467

order512
nilpotency class3
isoclinismisoclinic class of #468
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#21416

Subgroups

subgroupG/H
centerC2×C42C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C22⋊C4
prime radicalC2×#3444C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(11,13)(12,17,19,25)(14,22)(15,23)(16,21)(18,20,24,26);(4,7)(8,10)(13,21)(17,25)(20,26)(22,23);(3,5,6,9)(4,8,7,10)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6372471, #72317850
maximal normal subgroup ofC2×#6372471, #72317850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279663);