Finite Group Databasebeta

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Cayley diagram

C2×#54791 #8103736

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

Aliases

direct productC2×#54791

Subgroups

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

5892 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)(5,7)(6,9)(8,10)(12,15)(16,17)(19,20,25,26)(21,22,23,24);(4,5)(6,10)(11,12,14,15)(13,17,18,16)(19,21,25,23)(20,22,26,24);(4,5)(8,9)(11,13,14,18)(12,16,15,17)(19,22,25,24)(20,21,26,23);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,23,25,21)(20,22,26,24);(3,6,7,10)(4,8,5,9)(11,12)(13,16)(14,15)(17,18)(19,24,25,22)(20,21,26,23)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#8232279, C2×#9778754, C2×#8509798
maximal normal subgroup ofC2×#8232279, C2×#9778754, C2×#8509798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8010932);