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×#13234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1964 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
464 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)(6,7,12,16,13,17,11,29)(8,10,15,27,21,28,20,18)(9,22,25,31,26,36,24,14)(19,33,30,37,35,32,34,23);(6,8,18,7)(9,19,32,14)(10,16,11,20)(12,15,28,29)(13,21,27,17)(22,24,34,23)(25,30,37,36)(26,35,33,31);(6,9,12,25,13,26,11,24)(7,14,29,36,17,31,16,22)(8,19,15,30,21,35,20,34)(10,23,18,32,28,37,27,33)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6222);