Finite Group Databasebeta

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Cayley diagram

C2×#54981 #8098833

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

Aliases

direct productC2×#54981

Subgroups

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

5688 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)(5,7)(6,8)(9,10)(19,20)(21,23)(22,25)(24,26);(6,10)(8,9)(11,12,15,16)(13,14,17,18)(20,21)(24,25);(4,5)(8,9)(12,16)(13,17)(19,20,23,21)(22,24,26,25);(3,5,7,4)(6,9,10,8)(11,13)(12,14)(15,17)(16,18)(19,21,23,20)(22,25,26,24);(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,18)(16,17)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#9266284
maximal normal subgroup ofC2×#9266284

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8006029);