Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1961 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
466 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)(8,19)(9,24,28,33)(10,22)(11,16)(12,18)(13,17)(14,27,32,30)(15,20)(21,26,31,36)(23,29)(25,34,37,35);(6,8,20,18,11,23,29,16,12,15,19,7,13,22,10,17)(9,25,34,33,27,31,26,32,28,37,35,24,30,21,36,14);(6,9,12,28)(7,14)(8,21)(10,26,29,36)(11,27,13,30)(15,31)(16,24)(17,33)(18,32)(19,34,20,35)(22,37)(23,25)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6203);