Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C2×D4 |
| commutator | C4 | C22×C8×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C8×D4 |
| prime radical | C11×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C2×C11×C16 ◁ G | |
144 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(12,13,17,23,18,22,35,41,19,24,36,26,16,32,34,42)(14,25,28,30,27,20,37,31,21,38,40,29,33,39,43,15);(12,14)(13,20)(15,26)(16,33)(17,28)(18,27)(19,21)(22,38)(23,31)(24,39)(25,32)(29,41)(30,42)(34,43)(35,37)(36,40);(12,15,17,25,18,30,35,20,19,31,36,38,16,29,34,39)(13,21,23,40,22,33,41,43,24,14,26,28,32,27,42,37)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 903);