Finite Group Databasebeta

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Cayley diagram

C2×#39785 #7637400

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#39785

Subgroups

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

6460 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);(3,4,8,7)(5,10,6,9)(11,12,16,13)(14,17,15,18)(20,23)(24,26);(4,7)(9,10)(11,13,16,12)(14,17,15,18);(3,5,8,6)(4,9,7,10)(11,14,16,15)(12,17,13,18)(19,20)(21,24)(22,23)(25,26);(3,6,8,5)(4,10,7,9)(19,21,22,25)(20,24,23,26);(3,7,8,4)(5,10,6,9)(11,15,16,14)(12,18,13,17)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7544596);