Finite Group Databasebeta

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Cayley diagram

C2×#48637 #7703315

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#48637

Subgroups

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

6310 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);(12,16)(14,17)(19,20,24,22)(21,26,23,25);(3,4)(5,9)(6,10)(7,8)(11,12,15,16)(13,14,18,17)(20,22)(25,26);(4,8)(5,10)(11,13,15,18)(12,17,16,14)(19,21,24,23)(20,25,22,26);(3,5)(4,6)(7,10)(8,9)(11,13)(12,14)(15,18)(16,17)(19,22,24,20)(21,26,23,25);(3,6)(4,5)(7,9)(8,10)(11,14)(12,13)(15,17)(16,18)(19,23,24,21)(20,26,22,25)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7610511);