Finite Group Databasebeta

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Cayley diagram

C2×#57153 #8100794

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

Aliases

direct productC2×#57153

Subgroups

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

6210 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);(4,7)(8,10)(20,22)(21,23);(5,9)(8,10)(11,12,15,17)(13,14,18,16)(19,20)(21,25)(22,24)(23,26);(3,4,6,7)(5,8,9,10)(11,13,15,18)(12,16,17,14)(19,21,24,23)(20,25,22,26);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17)(19,22,24,20)(21,25,23,26);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,23,24,21)(20,25,22,26)

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, 8007990);