Finite Group Databasebeta

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Cayley diagram

C2×#38548 #7636034

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

Aliases

direct productC2×#38548

Subgroups

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

6112 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)(5,6)(7,8)(9,10)(19,20,23,24)(21,25,26,22);(4,8)(6,9)(12,15)(16,18)(21,26)(22,25);(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,26)(24,25);(3,5)(4,9)(6,8)(7,10)(11,13,14,17)(12,16,15,18);(3,6,7,9)(4,5,8,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,25)(24,26)

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, 7543230);