Finite Group Databasebeta

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Cayley diagram

C2×#55699 #8099211

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

Aliases

direct productC2×#55699

Subgroups

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

6340 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)(5,10)(6,9)(7,8)(12,15)(13,14)(19,20,22,24)(21,25,26,23);(3,5)(4,9)(6,8)(7,10)(11,12,16,15)(13,18,14,17)(21,26)(23,25);(5,6)(9,10)(11,13,16,14)(12,17,15,18)(19,21,22,26)(20,23,24,25);(3,6,8,5)(4,9,7,10)(11,14,16,13)(12,18,15,17)(19,20)(21,23)(22,24)(25,26);(3,7,8,4)(5,9,6,10)(11,15,16,12)(13,18,14,17)

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, 8006407);