Finite Group Databasebeta

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Cayley diagram

C2×#53949 #8098836

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

Aliases

direct productC2×#53949

Subgroups

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

5962 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);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,17)(16,18)(20,23)(21,25);(5,9)(8,10)(12,13)(16,17)(20,23)(24,26);(4,7)(8,10)(11,12,15,13)(14,16,18,17)(19,20)(21,24)(22,23)(25,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,25)(23,26);(3,5)(4,8)(6,9)(7,10)(11,14)(12,16)(13,17)(15,18)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10320757
maximal normal subgroup ofC2×#10320757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8006032);