Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13030 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1879 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
476 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)(6,7,10,15,11,17,27,31,12,18,9,16,26,35,24,13)(8,19,21,32,22,34,28,33,23,30,20,37,29,36,25,14);(7,13,18,31)(9,24)(10,27)(11,26)(14,30,33,19)(15,17,16,35)(20,25)(21,28)(22,29)(32,34,37,36);(6,8)(7,14)(9,25)(10,28)(11,29)(12,23)(13,30)(15,34)(16,36)(17,37)(18,33)(19,31)(20,24)(21,27)(22,26)(32,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6018);