Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C23×C11 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C11×#117 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,25,29)(22,26,31,39)(23,33,36,48)(24,37,34,28)(27,41,42,35)(30,43,45,51)(32,47,44,40)(38,49,50,46);(13,16)(17,19)(20,22,25,31)(21,26,29,39)(23,30,36,45)(24,32,34,44)(27,38,42,50)(28,40,37,47)(33,43,48,51)(35,46,41,49);(20,23,34,41,25,36,24,35)(21,27,37,48,29,42,28,33)(22,30,44,49,31,45,32,46)(26,38,47,51,39,50,40,43);(12,14,15,18)(13,17,16,19)(22,31)(26,39)(30,45)(32,44)(38,50)(40,47)(43,51)(46,49)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1909);