Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12807 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #410 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | #1799 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
428 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)(8,17)(9,21)(10,15,24,13)(11,16)(12,18)(14,23)(19,31,35,30)(20,32)(22,28,36,29)(25,33,26,37)(27,34);(6,8,9,18,11,20,23,34)(7,12,14,17,16,27,21,32)(10,19,22,33,24,35,36,37)(13,25,28,30,15,26,29,31);(7,13)(10,24)(12,25)(14,28)(15,16)(17,30)(19,35)(21,29)(22,36)(26,27)(31,32)(33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5795);