Finite Group Databasebeta

← Prev (#7637597) Next (#7637599) →

Cayley diagram

C2×#40112 #7637598

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#40112

Subgroups

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

6476 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)(5,9)(6,7)(8,10)(11,12,15,17)(13,14,18,16);(4,7)(8,9)(11,13,15,18)(12,16,17,14)(20,24)(22,25);(11,14)(12,13)(15,16)(17,18)(19,20)(21,25)(22,26)(23,24);(3,5,6,10)(4,8,7,9)(11,14)(12,13)(15,16)(17,18)(19,21,23,26)(20,25,24,22);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17)(19,22,23,25)(20,26,24,21)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7544794);