Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#2532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C11 |
| prime radical | C24×C11 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,14)(15,18,19,17)(21,22)(23,27)(30,34)(33,35);(13,14)(17,18)(20,21,24,22)(23,25,27,26)(28,29,32,31)(30,33,34,35);(12,14,16,13)(15,18,19,17)(20,22,24,21)(23,26,27,25)(28,30)(29,33)(31,35)(32,34);(12,15,16,19)(13,17,14,18)(20,23)(21,25)(22,26)(24,27)(28,31,32,29)(30,35,34,33)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1264);