Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#10426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(7,8)(12,17)(14,16)(18,19)(20,24)(21,25)(22,26)(23,27)(28,31)(29,32)(30,33);(6,7)(8,9)(10,11)(12,14)(13,15)(16,17)(18,20,23,30)(19,24,27,33)(21,29,22,28)(25,32,26,31);(6,8,9,7)(11,14)(12,17)(15,16)(19,25)(20,28)(21,22)(26,27)(29,30)(31,32)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3414);