Finite Group Databasebeta

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Cayley diagram

C2×#15085 #608215

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

Aliases

direct productC2×#15085

Subgroups

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

3662 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);(4,5)(8,10)(11,12,15,13)(14,16,18,17)(20,23)(24,26);(3,4,6,5)(12,13)(16,17)(19,20)(21,24)(22,23)(25,26);(7,8)(9,10)(11,13,15,12)(14,17,18,16)(19,20)(21,24)(22,23)(25,26);(3,5,6,4)(11,14,15,18)(12,16,13,17)(19,21,22,25)(20,24,23,26)

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, 515411);