Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22284 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2328 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3154 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
796 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)(8,10)(9,22,26,32)(11,17)(12,18)(13,20)(14,25,33,36)(15,27)(16,29)(19,34,35,30)(21,28)(23,24,31,37);(6,8,11,20,12,21,29,15)(7,13,16,10,18,27,17,28)(9,23,36,19,26,31,25,35)(14,24,22,30,33,37,32,34);(6,9,12,26)(7,14)(8,19)(10,24,28,37)(11,25,29,36)(13,30,27,34)(15,31)(16,32)(17,22)(18,33)(20,23)(21,35);(6,10,11,27,12,28,29,13)(7,15,16,8,18,20,17,21)(9,24,25,34,26,37,36,30)(14,31,32,19,33,23,22,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15272);