Finite Group Databasebeta

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Cayley diagram

C2×#29536 #6461811

order512
nilpotency class3
isoclinismisoclinic class of #23918
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#29536

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6501553, #72315997, #72316029, …
maximal normal subgroup ofC2×#6501553, #72315997, #72316029, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369007);