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×#34421 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1974 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)(12,15)(13,17)(14,16)(18,19,23,21)(20,24,28,26)(22,29,30,25)(27,32,33,31);(10,12,14,17)(11,15,16,13)(19,21)(24,26)(25,29)(31,32);(6,8,9,7)(10,12)(11,13)(14,17)(15,16)(18,20,23,28)(19,24,21,26)(22,27,30,33)(25,31,29,32);(10,13)(11,12)(14,15)(16,17)(18,21,23,19)(20,26,28,24)(22,29,30,25)(27,32,33,31);(18,22,23,30)(19,25,21,29)(20,27,28,33)(24,31,26,32)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27409);