Finite Group Databasebeta

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Cayley diagram

C2×#45281 #7669832

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#45281

Subgroups

subgroupG/H
center, socle, prime radicalC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

5018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

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

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

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, 7577028);