Finite Group Databasebeta

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Cayley diagram

C2×#7633 #160749

order512
nilpotency class3
isoclinismisoclinic class of #7606
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#7633

Subgroups

subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72217702, #72217722
maximal normal subgroup of#72217702, #72217722
maximal quotient ofC2×#96738, C2×#124891

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67945);