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Cayley diagram

C2×#19932 #6358302

order512
nilpotency class3
isoclinismisoclinic class of #19883
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#19932

Subgroups

subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,8,6)(5,11,12,18)(7,15,17,10)(9,16,14,13)(19,20,23,26)(21,27,29,34)(22,30,32,25)(24,31,33,28);(3,5,8,12)(4,9,6,14)(7,16,17,13)(10,18,15,11);(5,12)(9,14)(11,18)(13,16)(19,21,23,29)(20,24,26,33)(22,31,32,28)(25,34,30,27);(3,6,8,4)(5,11,12,18)(7,10,17,15)(9,16,14,13)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265498);