Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11334 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #392 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1444 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
652 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)(6,7,11,15,12,16,10,26)(8,17,21,28,20,32,19,13)(9,14,25,29,24,30,23,36)(18,33,35,27,22,31,34,37);(6,8,10,19,12,20,11,21)(7,13,15,17,16,28,26,32)(9,22,23,35,24,18,25,34)(14,27,29,31,30,37,36,33);(6,9)(7,14)(8,18,20,22)(10,23)(11,25)(12,24)(13,27,28,37)(15,29)(16,30)(17,31,32,33)(19,34,21,35)(26,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4322);