Finite Group Databasebeta

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Cayley diagram

C2×#16594 #611170

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#16594

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

3518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 518366);