Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17700 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | 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 ◁ C42×C5 ◁ C22×C42×C5 ◁ G | |
1016 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)(6,7,10,15,11,14,9,13)(8,16,17,21,18,12,19,20)(22,23)(24,31)(25,29)(26,34)(27,30)(28,32)(33,36)(35,37);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(22,24)(23,28)(25,32)(26,35)(27,33)(29,36)(30,37)(31,34);(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31);(6,9,11,10)(7,13,14,15)(8,17,18,19)(12,20,16,21)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10688);