Finite Group Databasebeta

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Cayley diagram

C2×#57177 #8100554

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

Aliases

direct productC2×#57177

Subgroups

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

5876 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)(5,7)(6,9)(8,10)(19,20,22,24)(21,25,26,23);(4,5)(8,9)(11,12,15,17)(13,14,18,16)(19,20,22,24)(21,23,26,25);(3,4,7,5)(6,8,10,9)(11,13,15,18)(12,16,17,14)(19,20)(21,23)(22,24)(25,26);(3,5,7,4)(6,9,10,8)(11,13)(12,14)(15,18)(16,17)(19,21)(20,23)(22,26)(24,25);(3,6,7,10)(4,8,5,9)(11,14)(12,13)(15,16)(17,18)

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, 8007750);