Finite Group Databasebeta

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Cayley diagram

C2×#47082 #7701055

order512
nilpotency class2
isoclinismisoclinic class of #46350
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#47082

Subgroups

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

6956 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);(3,4,7,5)(6,8,10,9)(11,12)(13,18)(14,17)(15,16)(20,22)(24,25);(3,4,7,5)(6,9,10,8)(11,13)(12,17)(14,16)(15,18)(19,20,23,22)(21,25,26,24);(4,5)(8,9)(13,14)(17,18);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,17,15,18)(19,21,23,26)(20,24,22,25);(3,6)(4,8)(5,9)(7,10)(11,15,16,12)(13,17,14,18)(19,22,23,20)(21,25,26,24)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7851893
maximal normal subgroup ofC2×#7851893

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7608251);