Finite Group Databasebeta

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Cayley diagram

C2×#59869 #7667024

order512
nilpotency class2
isoclinismisoclinic class of #58990
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C211×C4

Aliases

direct productC2×#59869

Subgroups

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

5350 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);(3,4,7,5)(6,9,10,8)(12,15)(13,17)(19,20,24,21)(22,26,23,25);(4,5)(8,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,20)(22,25,23,26);(19,22,24,23)(20,25,21,26);(3,5,7,4)(6,9,10,8)(19,23,24,22)(20,26,21,25);(3,6,7,10)(4,8,5,9)(11,13,14,17)(12,16,15,18)

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, 7574220);