Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#25206 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
858 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)(6,7,9,8)(10,11)(12,15)(13,14)(16,17)(18,19)(20,26)(21,24)(22,25)(23,28)(27,32)(29,31)(30,33);(6,8,9,7)(11,14)(15,17)(18,20,21,28)(19,23,24,26)(22,27,30,29)(25,31,33,32);(7,8)(10,12,13,16)(11,15,14,17)(20,27)(22,30)(23,31)(25,33)(26,32)(28,29);(11,14)(15,17)(19,24)(23,26)(25,33)(31,32)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18194);