Finite Group Databasebeta

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Cayley diagram

C2×#36900 #7634138

order512
nilpotency class2
isoclinismisoclinic class of #36702
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#36900

Subgroups

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

5678 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);(5,9)(8,10)(11,12,14,16)(13,15,17,18)(20,22)(21,25);(4,7)(8,10)(19,20,23,22)(21,26,25,24);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,25)(20,24,22,26);(3,4)(5,8)(6,7)(9,10)(19,22,23,20)(21,26,25,24);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,15,16,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7541334);