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×#26796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
838 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)(7,8)(10,11)(12,17)(13,15)(14,16)(18,19,22,25)(20,23,28,32)(21,24,29,33)(26,30,27,31);(6,7,9,8)(10,12,14,13)(11,15,16,17)(18,20,22,28)(19,23,25,32)(21,26,29,27)(24,30,33,31);(6,7,9,8)(10,13,14,12)(11,15,16,17)(20,26)(21,29)(23,30)(24,33)(27,28)(31,32);(6,8,9,7)(11,16)(15,17)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19784);