Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12982 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1869 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
474 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)(6,7,11,17,12,18,30,36,13,16,31,35,10,15,20,33)(8,19,23,28,21,34,37,9,22,25,29,24,27,14,32,26);(6,8,20,37,10,27,31,23,13,22,30,32,12,21,11,29)(7,14,33,28,15,25,35,26,16,34,36,24,18,19,17,9);(6,9)(7,8)(10,28)(11,19)(12,24)(13,26)(14,20)(15,27)(16,22)(17,29)(18,21)(23,35)(25,31)(30,34)(32,36)(33,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5970);