Finite Group Databasebeta

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Cayley diagram

C2×#59927 #7666793

order512
nilpotency class2
isoclinismisoclinic class of #58990
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#59927

Subgroups

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

4982 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);(5,9)(6,10)(11,12,15,14)(13,17,18,16);(3,4,8,7)(5,10,9,6)(11,13)(12,16)(14,17)(15,18)(19,20,22,24)(21,25,26,23);(3,5)(4,6)(7,10)(8,9)(11,14,15,12)(13,17,18,16);(3,6,8,10)(4,9,7,5)(19,21,22,26)(20,23,24,25);(3,7,8,4)(5,10,9,6)(11,13)(12,16)(14,17)(15,18)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573989);