Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#21244 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#390 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1460 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)(8,12,17,26)(9,19)(10,22)(11,14)(13,21)(15,23)(16,28,29,37)(18,32,31,25)(20,33)(24,35,36,30)(27,34);(6,8)(7,12,14,26)(9,16)(10,18)(11,17)(13,24,27,36)(15,25,22,32)(19,28,33,37)(20,29)(21,30)(23,31)(34,35);(6,9,10,21,11,20,23,34)(7,13,15,19,14,27,22,33)(8,16,18,30,17,29,31,35)(12,24,25,28,26,36,32,37);(9,20)(13,27)(16,29)(19,33)(21,34)(24,36)(28,37)(30,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14232);