Finite Group Databasebeta

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Cayley diagram

C2×#15080 #608195

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#15080

Subgroups

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

3726 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);(8,10)(11,12,14,16)(13,15,17,18)(19,20,23,26)(21,25,22,24);(4,6)(7,8)(9,10)(19,21,23,22)(20,24,26,25);(3,4,5,6)(11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,25)(23,26);(11,13,14,17)(12,15,16,18)(19,22,23,21)(20,25,26,24)

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, 515391);