Finite Group Databasebeta

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Cayley diagram

C2×#13539 #499549

order512
nilpotency class4
isoclinismisoclinic class of #13304
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#13539

Subgroups

subgroupG/H
center, socle, prime radicalC24C2×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,7,9)(5,10,6,8)(11,12)(13,15)(14,16)(17,18)(19,20,22,24)(21,25,28,32)(23,26,30,31)(27,33,34,29);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18)(19,21,22,28)(20,25,24,32)(23,27,30,34)(26,33,31,29);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(20,24)(21,27)(23,31)(25,29)(26,30)(28,34)(32,33)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499581, C2×#511415, C2×#510848
maximal normal subgroup ofC2×#499581, C2×#511415, C2×#510848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406745);