Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×Q8 | C24×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×D4 ◁ C2×C5×2−1+4 ◁ G | |
1208 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)(6,7,10,15,11,16,23,30)(8,13,19,28,17,26,32,36)(9,14,21,29,20,12,22,25)(18,27,33,37,31,24,34,35);(7,12)(9,20)(13,24)(14,16)(15,29)(18,31)(21,22)(25,30)(26,27)(28,37)(33,34)(35,36);(6,8)(7,13)(9,18)(10,19)(11,17)(12,24)(14,27)(15,28)(16,26)(20,31)(21,33)(22,34)(23,32)(25,35)(29,37)(30,36);(8,17)(13,26)(18,31)(19,32)(24,27)(28,36)(33,34)(35,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26358);