Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#12011 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C2×D8 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C2×C5 | C2×#335 |
| prime radical | C23×C5 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
220 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)(6,7,10,15,11,17,9,25,12,18,20,14,26,16,27,32)(8,19,23,30,24,34,22,13,21,29,35,31,37,33,36,28);(6,8,20,35,11,24,27,36,12,21,10,23,26,37,9,22)(7,13,25,33,16,30,15,29,18,28,32,34,17,31,14,19);(8,21)(13,28)(19,29)(22,36)(23,35)(24,37)(30,31)(33,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4999);