Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8986 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
532 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)(10,11,15,16,14,12,13,17)(18,19)(20,26,28,33)(21,23)(22,25)(24,27,31,30)(29,32);(6,7,8,9)(10,12,14,11)(13,17,15,16)(19,23)(22,29)(24,31)(27,30);(10,13,14,15)(11,16,12,17)(18,20)(19,24)(21,28)(22,30)(23,31)(25,33)(26,32)(27,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1974);