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Cayley diagram

C2×#28698 #6466266

order512
nilpotency class3
isoclinismisoclinic class of #24390
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28698

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1934 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)(15,16,18,21)(17,23,25,30)(19,28,27,22)(20,29,24,26);(3,4,5,6)(7,9,11,10)(8,12,14,13)(15,16)(17,24)(18,21)(19,22)(20,25)(23,29)(26,30)(27,28);(4,6)(15,17,19,26,18,25,27,29)(16,20,22,23,21,24,28,30);(7,10,11,9)(8,13,14,12)(16,21)(20,24)(22,28)(23,30)

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, 6373462);