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Cayley diagram

C2×#39619 #7637287

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#39619

Subgroups

subgroupG/H
center, socle, prime radicalC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 30 points:

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

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, 7544483);