Finite Group Databasebeta

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Cayley diagram

C2×#55104 #8101334

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

Aliases

direct productC2×#55104

Subgroups

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

5764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(1,2);(11,12,14,15)(13,17,18,16)(19,20)(21,25)(22,24)(23,26);(3,4,7,8)(5,6,9,10)(12,15)(16,17)(19,21,24,23)(20,25,22,26);(4,8)(5,9)(11,12)(13,16)(14,15)(17,18)(19,22,24,20)(21,25,23,26);(3,5)(4,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,23,24,21)(20,25,22,26);(3,6)(4,5)(7,10)(8,9)(11,13)(12,16)(14,18)(15,17)

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, 8008530);