Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2399 |
| 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 | |
504 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 1408 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(16,17,20,23)(18,22,19,21)(24,25,28,31)(26,30,27,29);(12,13)(14,15)(16,18,20,19)(17,21,23,22)(24,26,28,27)(25,29,31,30);(13,15)(16,18,20,19)(17,22,23,21)(24,27,28,26)(25,30,31,29);(16,19,20,18)(17,21,23,22)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1131);