Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12618 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1752 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×M32 ◁ G | |
382 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)(6,7,9,14,10,15,20,28,11,12,19,23,21,24,33,35)(8,13,17,26,16,27,29,37,18,22,32,34,30,25,31,36);(7,12)(8,16,18,30)(9,19)(13,25,22,27)(15,24)(17,31,32,29)(20,33)(26,37,34,36);(6,8,9,17,10,16,20,29,11,18,19,32,21,30,33,31)(7,13,14,26,15,27,28,37,12,22,23,34,24,25,35,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5606);