Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21558 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C4×#121 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,12,17,13,18,11,28)(8,19,10,16,22,33,21,14)(9,23,26,15,27,31,25,32)(20,29,24,34,36,30,35,37);(6,8,11,21,13,22,12,10)(7,14,17,19,18,16,28,33)(9,20,25,35,27,36,26,24)(15,29,31,34,32,30,23,37);(6,9)(7,15)(8,20)(10,24)(11,25)(12,26)(13,27)(14,29)(16,30)(17,31)(18,32)(19,34)(21,35)(22,36)(23,28)(33,37);(6,10,12,22,13,21,11,8)(7,16,17,33,18,14,28,19)(9,24,26,36,27,35,25,20)(15,30,31,37,32,29,23,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14546);