Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1778 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D4 |
| commutator | C4 | C2×C42×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C22×C11 | C2×C4⋊C4 |
| prime radical | C23×C11 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,14)(16,17,20,25,21,23,19,24)(18,26,28,22,29,31,27,30);(16,18,19,28,21,29,20,27)(17,22,24,31,23,30,25,26);(12,14,15,13)(17,23)(22,30)(24,25)(26,31)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 510);