Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#22039 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C5×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1028 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)(8,9)(10,11)(12,16)(13,15)(14,17)(18,19,22,26,23,20,24,28)(21,25,30,32,29,27,31,33);(10,12)(11,14)(13,17)(15,16)(18,20,22,28,23,19,24,26)(21,27,30,33,29,25,31,32);(7,9)(12,17)(14,16)(18,21)(19,25)(20,27)(22,30)(23,29)(24,31)(26,32)(28,33);(11,15)(14,16)(19,20)(21,29)(26,28)(30,31)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15027);