Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C2×D8 |
| commutator | C8 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C4×D8 |
| prime radical | C11×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,22,18,24,34,23)(14,25,27,40,29,42,28,36)(15,21,30,39,31,20,16,32)(19,35,37,43,38,33,41,26);(12,14)(13,19)(15,26)(16,33)(17,27)(18,29)(20,36)(21,40)(22,37)(23,41)(24,38)(25,32)(28,34)(30,35)(31,43)(39,42);(12,15,17,30,18,31,34,16)(13,20,22,32,24,21,23,39)(14,26,27,35,29,43,28,33)(19,36,37,25,38,40,41,42)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 911);