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Cayley diagram

C2×#25476 #6446147

order512
nilpotency class3
isoclinismisoclinic class of #24269
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#25476

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353343);