Finite Group Databasebeta

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Cayley diagram

C2×#15242 #608580

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#15242

Subgroups

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

3966 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)(8,10)(11,12)(13,15)(14,17)(16,18)(19,20,24,21)(22,26,23,25);(4,6)(7,8,9,10)(12,13)(16,17)(19,21,24,20)(22,26,23,25);(3,4,5,6)(11,13,15,12)(14,17,18,16)(19,22,24,23)(20,25,21,26);(11,14,15,18)(12,16,13,17)(19,23,24,22)(20,26,21,25)

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, 515776);