Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#120 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
1060 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)(6,7,9,14,11,15,21,29)(8,13,17,27,19,28,32,37)(10,12,20,24,22,25,33,35)(16,26,30,36,18,23,31,34);(7,12)(9,20)(13,23)(15,25)(17,31)(21,33)(26,28)(30,32);(6,8,10,18)(7,13,12,23)(9,17,20,31)(11,19,22,16)(14,27,24,34)(15,28,25,26)(21,32,33,30)(29,37,35,36);(8,16)(13,26)(17,30)(18,19)(23,28)(27,36)(31,32)(34,37)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10410);