Finite Group Databasebeta

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Cayley diagram

C2×#53677 #8103388

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

Aliases

direct productC2×#53677

Subgroups

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

5946 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);(11,12)(13,17)(14,16)(15,18)(19,20,22,24)(21,25,26,23);(3,4)(5,9)(6,7)(8,10)(11,12,14,16)(13,15,18,17)(19,20,22,24)(21,23,26,25);(4,7)(8,9)(19,20)(21,23)(22,24)(25,26);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,22,26)(20,23,24,25);(3,5)(4,8)(6,10)(7,9)(11,13)(12,15)(14,18)(16,17)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#9778754, C2×#10087048, C2×#8471393
maximal normal subgroup ofC2×#9778754, C2×#10087048, C2×#8471393

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8010584);