Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17716 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C5×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C42×C5 ◁ G | |
994 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)(6,7,10,13,11,14,9,18)(8,15,16,20,17,12,19,21)(22,23)(24,32)(25,30)(26,29)(27,31)(28,33)(34,36)(35,37);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,15)(18,21)(22,24)(23,28)(25,33)(26,35)(27,34)(29,32)(30,36)(31,37);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,15,21)(22,25,27,26)(23,29,31,30)(24,33,34,35)(28,32,37,36);(22,25,27,26)(23,30,31,29)(24,33,34,35)(28,36,37,32)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10704);