Finite Group Databasebeta

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Cayley diagram

C2×#42065 #7640366

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#42065

Subgroups

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

6882 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)(5,9)(6,8)(7,10)(19,20)(21,26)(22,24)(23,25);(3,4,6,8)(5,7,10,9)(11,12)(13,14)(15,16)(17,18)(19,20,24,22)(21,25,23,26);(5,10)(7,9)(12,16)(13,17)(19,21,24,23)(20,25,22,26);(3,5,6,10)(4,7,8,9)(11,13,15,17)(12,14,16,18)(19,22,24,20)(21,25,23,26);(3,4)(5,7)(6,8)(9,10)(11,14)(12,17)(13,16)(15,18)(19,23,24,21)(20,25,22,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10442541
maximal normal subgroup ofC2×#10442541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7547562);