Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
944 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)(7,12,14,26)(9,20)(13,25,29,36)(15,27,16,21)(18,33)(22,24)(28,34,30,37)(31,35);(6,7,10,15)(8,13,19,30)(9,21,20,26)(11,16,23,14)(12,24,27,22)(17,28,32,29)(18,34,33,36)(25,35,37,31);(6,8,10,19)(7,13,15,30)(9,18,20,33)(11,17,23,32)(12,25,27,37)(14,29,16,28)(21,34,26,36)(22,31,24,35);(8,17)(13,28)(18,31)(19,32)(25,34)(29,30)(33,35)(36,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10597);