Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#1764 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | D4 |
| commutator | C4 | C2×C42×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C22×C11 | C4×D4 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ C22×C8×C11 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,20,24,21,25,19,22)(18,23,27,30,28,31,26,29);(13,15)(17,22,25,24)(23,29,31,30);(16,18,20,27,21,28,19,26)(17,23,24,30,25,31,22,29)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 496);