Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#9733 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(7,9)(10,11,12,15,13,16,17,14)(18,19)(20,25)(21,30)(22,32)(23,27)(24,28)(26,31)(29,33);(6,7,8,9)(11,14)(12,17)(15,16)(18,20,22,28,23,29,31,21)(19,24,26,25,27,30,32,33);(18,21,31,29,23,28,22,20)(19,25,32,24,27,33,26,30)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2721);