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×#13103 |
| 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 | |
646 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)(6,7,12,17,13,18,11,16)(8,10,15,28,21,29,30,19)(9,22,25,14,26,32,24,33)(20,35,31,23,36,37,34,27);(6,8,19,7)(9,20,35,14)(10,17,11,30)(12,15,29,16)(13,21,28,18)(22,25,31,27)(23,32,24,34)(26,36,37,33);(6,9,13,26)(7,14,18,33)(8,20,21,36)(10,27,29,23)(11,25,12,24)(15,34,30,31)(16,32,17,22)(19,35,28,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6091);