Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7258 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #102 |
| commutator | C2×C8 | C42×C5 |
| Frattini, cosocle | C23×C8 | C22×C5 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,14)(13,16)(15,17)(18,19)(20,23)(21,30)(22,26)(24,27)(25,28)(29,32)(31,33);(11,14,16,15)(19,23)(20,27)(21,28)(24,25)(26,32)(29,33)(30,31)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 246);