Finite Group Databasebeta

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Cayley diagram

C2×#14794 #607731

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

Aliases

direct productC2×#14794

Subgroups

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

3846 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)(11,12)(13,14)(15,16)(17,18)(19,20,24,22)(21,26,23,25);(8,10)(12,16)(13,18)(19,21,24,23)(20,25,22,26);(7,8)(9,10)(11,13)(12,17)(14,16)(15,18)(19,22,24,20)(21,26,23,25);(3,4)(5,6)(11,14,15,17)(12,13,16,18)(19,23,24,21)(20,26,22,25)

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, 514927);