Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#120 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1046 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)(6,7,9,14,11,15,20,27)(8,13,17,26,16,25,28,35)(10,12,19,23,21,24,31,34)(18,22,30,33,29,32,36,37);(7,12)(9,19)(13,22)(15,24)(17,30)(20,31)(25,32)(28,36);(6,8,10,18)(7,13,12,22)(9,17,19,30)(11,16,21,29)(14,26,23,33)(15,25,24,32)(20,28,31,36)(27,35,34,37);(8,16)(13,25)(17,28)(18,29)(22,32)(26,35)(30,36)(33,37)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10400);