Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1874 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,12,18,27,30,13,19,9,25,10,17,26,14)(8,20,23,29,22,33,37,34,24,35,21,36,28,15,31,32);(7,14,19,30)(9,26)(10,12)(11,27)(15,29,33,36)(16,18,25,17)(20,32,35,34)(21,31)(22,28)(23,37);(6,8)(7,15)(9,23)(10,28)(11,21)(12,22)(13,24)(14,29)(16,34)(17,35)(18,20)(19,33)(25,32)(26,37)(27,31)(30,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5993);