Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23730 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C22×D4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
1126 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)(10,11,13,15)(12,16,17,14)(18,19,22,27,23,28,21,26)(20,29,30,25,32,33,31,24);(6,7)(8,9)(10,12,13,17)(11,14,15,16)(18,20)(19,24)(21,30)(22,31)(23,32)(25,28)(26,29)(27,33);(7,9)(18,20)(19,25)(21,30)(22,31)(23,32)(24,28)(26,33)(27,29);(18,21,23,22)(19,26,28,27)(20,30,32,31)(24,29,25,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16718);