Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#35318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2106 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)(7,9)(11,15)(12,17)(18,19,23,21)(20,24,26,27)(22,29,30,25)(28,33,32,31);(10,11)(12,16)(13,17)(14,15)(19,21)(20,26)(25,29)(28,32);(6,7,8,9)(10,12,14,17)(11,16,15,13)(18,20)(19,24)(21,27)(22,28)(23,26)(25,31)(29,33)(30,32);(10,13,14,16)(11,17,15,12)(18,21,23,19)(20,27,26,24)(22,29,30,25)(28,33,32,31);(18,22,23,30)(19,25,21,29)(20,28,26,32)(24,31,27,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28306);