Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33428 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#432 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C5×#405 ◁ G | |
528 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)(6,7,9,15,10,16,22,33,11,14,23,29,24,30,36,37)(8,17,19,31,20,27,35,32,18,12,21,25,34,13,26,28);(6,8)(7,12)(9,21)(10,20)(11,18)(13,16)(14,17)(15,31)(19,23)(22,26)(24,34)(25,29)(27,30)(28,37)(32,33)(35,36);(6,8)(7,13,14,27)(9,21)(10,20)(11,18)(12,16,17,30)(15,32,29,28)(19,23)(22,26)(24,34)(25,37,31,33)(35,36);(8,18)(12,17)(13,27)(19,21)(20,34)(25,31)(26,35)(28,32)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26416);