Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#34487 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(10,11,14,12)(13,16,17,15)(19,23)(20,26)(21,29)(31,33);(11,12)(15,16)(18,19,22,23)(20,27,26,24)(21,25,29,30)(28,33,32,31);(18,20,22,26)(19,24,23,27)(21,28,29,32)(25,31,30,33);(6,8,9,7)(10,12,14,11)(13,16,17,15)(18,19)(20,24)(21,25)(22,23)(26,27)(28,31)(29,30)(32,33);(10,13,14,17)(11,15,12,16)(18,21,22,29)(19,25,23,30)(20,28,26,32)(24,31,27,33)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27475);