Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33463 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C2×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×C4×D8 |
| prime radical | C5×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
652 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)(6,7,11,16,12,15,27,17)(8,18,20,29,22,35,21,37)(9,14,23,33,24,32,10,25)(13,28,31,36,30,26,34,19);(6,8)(7,13)(9,19)(10,26)(11,20)(12,22)(14,29)(15,30)(16,31)(17,34)(18,25)(21,27)(23,28)(24,36)(32,37)(33,35);(6,9,11,23,12,24,27,10)(7,14,16,33,15,32,17,25)(8,19,20,28,22,36,21,26)(13,29,31,35,30,37,34,18);(7,15)(13,30)(14,32)(16,17)(18,35)(25,33)(29,37)(31,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26451);