Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#21989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(6,7,8,9)(11,14)(12,16)(18,19)(20,27)(21,29)(22,26)(23,24)(25,30)(28,33)(31,32);(10,11)(12,15)(13,14)(16,17)(18,20,22,23)(19,24,26,27)(21,30,31,28)(25,29,33,32);(11,14)(15,17)(18,21,23,30,22,31,20,28)(19,25,27,29,26,33,24,32);(10,12,13,16)(11,15,14,17)(19,26)(21,31)(24,27)(28,30)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14977);