Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#12577 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | C2×#342 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,20,21,13)(10,14)(11,15)(12,18,24,30)(17,28)(19,29)(22,33,34,26)(23,27)(25,32,35,37)(31,36);(6,8,10,19,11,17,23,31)(7,12,14,25,15,24,27,35)(9,18,22,32,21,30,34,37)(13,16,26,29,20,28,33,36);(8,17)(12,24)(16,28)(18,30)(19,31)(25,35)(29,36)(32,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5565);