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Cayley diagram

C2×#25471 #6446141

order512
nilpotency class3
isoclinismisoclinic class of #24264
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#25471

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

2130 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)(15,16,19,24,21,26,20,25)(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)(16,22,24,23,26,29,25,30);(4,6)(15,18,21,17)(16,23,26,30)(19,27,20,28)(22,25,29,24);(7,10,11,9)(8,13,14,12)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353337);