Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13229 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1964 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
452 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)(8,18,22,33)(9,23)(10,26,19,15)(11,29)(12,17)(13,14)(16,30)(20,34,35,28)(21,24,36,32)(25,37)(27,31);(6,8,19,14,12,22,10,27)(7,13,26,33,17,31,15,18)(9,20,32,16,25,35,24,29)(11,21,34,23,30,36,28,37);(6,9,11,13,12,25,30,31)(7,14,16,23,17,27,29,37)(8,20,21,26,22,35,36,15)(10,24,28,18,19,32,34,33)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6217);