Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2402 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C11 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C11 |
| socle, prime radical | C23×C11 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C11 ◁ G | |
400 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 400 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) = 35 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,27)(24,26)(28,29,31,33)(30,32,34,35);(13,16)(17,19)(20,22,24,23)(21,25,26,27)(28,30)(29,32)(31,34)(33,35);(20,23,24,22)(21,25,26,27)(30,34)(32,35);(12,14,15,18)(13,17,16,19)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1134);