Finite Group Databasebeta

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Cayley diagram

C2×#53165 #8105650

order512
nilpotency class2
isoclinismisoclinic class of #53158
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#53165

Subgroups

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

6752 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);(5,9)(8,10)(12,16)(13,18)(20,23)(24,26);(4,7)(8,10)(11,12,15,16)(13,17,18,14)(21,25)(24,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,18)(12,17,16,14)(19,20)(21,24)(22,23)(25,26);(3,5)(4,8)(6,9)(7,10)(11,14,15,17)(12,18,16,13);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,25)(23,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012846);