Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1320 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #99 |
| commutator | C23 | C2×C8×C11 |
| Frattini, cosocle | C2×M16 | C22×C11 |
| socle | C2×C11 | #323 |
| prime radical | C24×C11 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C11×M16 ◁ G | |
128 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,18,15,19,21,25,16,17,20,23,22,24,26,27);(13,17)(19,24)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 52);