Finite Group Databasebeta

← Prev (#7640291) Next (#7640293) →

Cayley diagram

C2×#42130 #7640292

order512
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#42130

Subgroups

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

6270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

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, 7547488);