Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#538 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(27,33)(34,37);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(22,23)(24,27)(25,28)(26,29)(30,34)(31,33)(32,35)(36,37);(6,7)(8,12)(9,18)(10,13)(11,14)(15,16)(17,21)(19,20)(22,24)(23,27)(25,30)(26,31)(28,34)(29,33)(32,36)(35,37);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,25,26,32)(23,28,29,35)(24,30,31,36)(27,34,33,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11239);