Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13098 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1898 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×2−1+4 ◁ G | |
640 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)(8,10,18,25)(9,19,22,31)(11,14)(12,15)(13,21,30,27)(16,33)(17,24,26,35)(20,34)(23,36)(28,32)(29,37);(6,8,10,15,12,18,25,7)(9,16,23,30,22,29,20,13)(11,26,24,14,28,17,35,32)(19,27,34,37,31,21,36,33);(6,9)(7,13)(8,16)(10,23)(11,27)(12,22)(14,31)(15,30)(17,36)(18,29)(19,32)(20,25)(21,28)(24,37)(26,34)(33,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6086);