Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23042 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2361 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,13,17)(8,18)(9,23,25,10)(11,16,12,29)(14,28)(15,30,27,24)(19,20)(21,34)(22,33)(26,36)(31,37)(32,35);(6,8,12,21,13,22,11,28)(7,14,29,18,17,34,16,33)(9,19,30,36,25,37,24,32)(10,26,15,31,23,35,27,20);(6,9)(7,15)(8,19)(10,16)(11,24)(12,30)(13,25)(14,31)(17,27)(18,35)(20,34)(21,36)(22,37)(23,29)(26,33)(28,32);(6,10,11,27,13,23,12,15)(7,9,16,24,17,25,29,30)(8,20,21,26,22,31,28,35)(14,32,18,19,34,36,33,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16030);