Finite Group Databasebeta

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Cayley diagram

C2×#58374 #8104013

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#58374

Subgroups

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

5876 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);(11,12)(13,16)(14,15)(17,18)(19,20,24,25)(21,23,22,26);(3,4,7,5)(6,9,10,8)(12,14)(13,17)(20,25)(21,22);(4,5)(8,9)(11,13,15,17)(12,16,14,18)(19,21,24,22)(20,26,25,23);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,22,24,21)(20,23,25,26);(3,6)(4,8)(5,9)(7,10)(11,14,15,12)(13,18,17,16)(19,23)(20,21)(22,25)(24,26)

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, 8011209);