Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7316 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1304 |
| commutator | C24 | C42×C5 |
| Frattini, cosocle | C23×Q8 | C22×C5 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C23×C5×Q8 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,11,15,22,30)(8,16,17,28,19,12,20,26)(10,23,24,29,25,14,21,31)(18,27,34,37,35,32,33,36);(6,8,10,18,11,19,25,35)(7,12,14,27,15,16,23,32)(9,20,21,34,22,17,24,33)(13,28,29,37,30,26,31,36)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 305);