Finite Group Databasebeta

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Cayley diagram

C2×#51961 #7707883

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#51961

Subgroups

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

5576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

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, 7615079);