Finite Group Databasebeta

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Cayley diagram

C2×#38581 #7636039

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#38581

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

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

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, 7543235);