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×#26903 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G | |
836 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)(10,11,13,14)(12,16,17,15)(18,19,24,27,23,29,22,28)(20,30,33,25,21,32,31,26);(6,7)(8,9)(11,14)(15,16)(18,20,23,21)(19,25,29,26)(22,33,24,31)(27,30,28,32);(7,9)(18,21,23,20)(19,26,29,25)(22,31,24,33)(27,32,28,30);(10,12,13,17)(11,15,14,16)(19,27)(22,24)(25,30)(26,32)(28,29)(31,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19891);