Finite Group Databasebeta

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Cayley diagram

C2×#20728 #6370948

order512
nilpotency class3
isoclinismisoclinic class of #1462
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#20728

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C22⋊C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4450 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)(11,12,14,18,15,17,21,24)(13,16,19,23,20,22,25,26);(4,7)(8,10)(11,13,15,20)(12,16,17,22)(14,19,21,25)(18,23,24,26);(12,17)(14,21)(16,22)(19,25);(3,5,6,9)(4,8,7,10)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239475, #72239498, C2×#185759, …
maximal normal subgroup of#72239475, #72239498, C2×#185759, …
maximal quotient of#72239496, #72264416

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278144);