Finite Group Databasebeta

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Cayley diagram

C2×#52480 #8092167

order512
nilpotency class2
isoclinismisoclinic class of #52308
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#52480

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999363);