Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 704 |
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)(17,21)(18,25)(19,26)(22,27)(23,28)(24,31)(29,30);(16,17,22,31,19,24,18,21,20,23,25,30,26,29,27,28);(13,15)(17,23)(21,28)(24,29)(30,31)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 952);