Finite Group Databasebeta

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Cayley diagram

C2×#13507 #499402

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

Aliases

direct productC2×#13507

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

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499581, C2×#511530, C2×#510848
maximal normal subgroup ofC2×#499581, C2×#511530, C2×#510848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406598);