Finite Group Databasebeta

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Cayley diagram

C2×#57402 #8100583

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#57402

Subgroups

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

5796 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);(3,4,6,7)(5,9,10,8)(12,15)(16,18);(4,7)(5,10)(13,17)(16,18)(19,20,23,25)(21,22,26,24);(4,7)(8,9)(11,12,14,15)(13,16,17,18)(19,21,23,26)(20,24,25,22);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,26)(24,25);(3,5)(4,8)(6,10)(7,9)(11,13,14,17)(12,16,15,18)(19,22)(20,21)(23,24)(25,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8007779);