Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21184 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#389 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,15,12,16,10,26)(8,17,19,13,21,28,20,29)(9,22,25,30,23,36,24,14)(18,27,33,35,32,37,34,31);(6,8,11,19,12,21,10,20)(7,13,15,28,16,29,26,17)(9,18,25,33,23,32,24,34)(14,27,22,35,30,37,36,31);(6,9,10,24,12,23,11,25)(7,14,15,22,16,30,26,36)(8,18,20,34,21,32,19,33)(13,27,28,35,29,37,17,31);(9,23)(14,30)(18,32)(22,36)(24,25)(27,37)(31,35)(33,34)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14172);