Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23043 |
| 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 | |
820 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)(6,7,12,17)(8,18)(9,10,27,30)(11,15,19,16)(13,23)(14,25,29,26)(20,31)(21,37)(22,33)(24,34)(28,32)(35,36);(6,8,19,22,12,24,11,23)(7,13,16,18,17,33,15,34)(9,20,26,36,27,37,25,32)(10,28,14,31,30,35,29,21);(6,9)(7,14)(8,20)(10,15)(11,25)(12,27)(13,31)(16,30)(17,29)(18,35)(19,26)(21,33)(22,36)(23,32)(24,37)(28,34);(6,10,11,29,12,30,19,14)(7,9,15,25,17,27,16,26)(8,21,22,28,24,31,23,35)(13,32,18,20,33,36,34,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16031);