Finite Group Databasebeta

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Cayley diagram

C2×#19088 #6351706

order512
nilpotency class3
isoclinismisoclinic class of #19086
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#19088

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

2872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(4,9)(5,12)(6,7)(11,14)(13,15)(16,18)(19,20)(21,27)(22,30)(23,25)(24,29)(26,31)(28,34)(32,33);(3,4)(5,10)(6,14)(7,9)(8,11)(12,18)(13,17)(15,16)(22,31)(26,30)(27,34)(29,32);(3,5,8,13)(4,10,11,17)(6,12,7,15)(9,16,14,18)(19,21)(20,24)(22,32)(23,28)(25,33)(26,34)(27,30)(29,31);(4,11)(9,14)(10,17)(16,18)(20,25)(24,33)(26,30)(27,34)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258902);