Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#7217 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #340 |
| commutator | C2×C8 | C42×C5 |
| Frattini, cosocle | C2×C4×C8 | C22×C5 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14,16,20)(9,12)(10,13)(11,15,18,21)(17,19)(23,28)(24,32)(25,34)(26,33)(29,30)(31,37)(35,36);(6,8)(7,11,13,18)(9,15)(10,16)(12,14,19,20)(17,21)(22,23,25,29,26,30,24,28,27,31,32,36,33,35,34,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 205);