Finite Group Databasebeta

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Cayley diagram

C2×#47192 #7701010

order512
nilpotency class2
isoclinismisoclinic class of #46350
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#47192

Subgroups

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

6348 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)(5,9)(6,8)(7,10)(19,20,23,24)(21,25,26,22);(3,4,6,8)(5,7,10,9)(11,12,15,16)(13,14,17,18)(20,24)(22,25);(12,16)(13,17)(19,21)(20,22)(23,26)(24,25);(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,17)(16,18)(19,22)(20,21)(23,25)(24,26);(3,5,6,10)(4,7,8,9)(11,14)(12,13)(15,18)(16,17)(19,21)(20,22)(23,26)(24,25)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7608206);