Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#34189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2342 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)(6,7,8,9)(12,16)(14,17)(18,19)(20,24)(21,28)(22,23)(25,29)(26,32)(27,30)(31,33);(10,11,13,15)(12,14,16,17)(19,23)(24,30)(25,28)(31,32);(10,12,13,16)(11,14,15,17)(18,20,22,27)(19,24,23,30)(21,26,29,33)(25,31,28,32);(18,19)(20,24)(21,25)(22,23)(26,31)(27,30)(28,29)(32,33);(18,21,22,29)(19,25,23,28)(20,26,27,33)(24,31,30,32)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27177);