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) = C24 |
| direct product | C11×#1738 |
| 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 | C24×C11 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C23×C4×C11 ◁ G | |
372 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(16,17,19,24,21,25,20,23)(18,26,28,22,27,31,29,30);(13,15)(16,18,20,29,21,27,19,28)(17,22,24,31,25,30,23,26);(18,27)(22,30)(26,31)(28,29)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 470);