Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#1426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #367 |
| commutator | C16 | C2×C4×C11 |
| Frattini, cosocle | C2×C16 | C22×C11 |
| prime radical | C22×C11 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,18,20,31)(14,23)(15,27)(16,28)(19,34)(21,35,38,43)(22,41,39,33)(24,29)(25,37)(26,36)(30,42,40,32);(12,13,19,35,15,21,14,18,28,39,23,32,24,30,34,41,17,20,37,43,29,38,26,31,16,22,36,42,27,40,25,33)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 158);