Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13008 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #481 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1874 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
516 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)(6,7,10,15,11,16,27,33,12,14,9,22,24,31,25,35)(8,17,20,30,21,32,26,29,19,36,18,34,28,37,23,13);(6,8)(7,13,14,29)(9,23)(10,26)(11,28)(12,19)(15,32,22,37)(16,34,31,30)(17,35,36,33)(18,25)(20,27)(21,24);(7,14)(9,10)(13,29)(16,31)(18,20)(23,26)(25,27)(30,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5996);