Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C11 |
| socle | C2×C11 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,23)(15,21)(16,28)(17,32)(18,20)(19,25)(22,30)(24,40)(26,39)(27,37)(29,35)(31,42)(33,38)(34,41)(36,43);(12,14)(13,19)(15,24)(16,29)(17,33)(18,26)(20,34)(21,35)(22,38)(23,28)(25,30)(27,41)(31,43)(32,36)(37,40)(39,42);(13,20)(19,34)(21,32)(22,37)(23,39)(28,42)(35,36)(38,40);(12,15,18,17)(13,21,20,32)(14,24,26,33)(16,27,31,30)(19,35,34,36)(22,28,37,42)(23,40,39,38)(25,29,41,43)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2146);