Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11310 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #387 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1443 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
532 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)(6,7,11,17,12,15,10,16)(8,18,20,33,21,13,26,28)(9,14,25,32,23,30,24,31)(19,34,22,37,35,27,36,29);(6,8)(7,13)(9,22,23,36)(10,20)(11,26)(12,21)(14,29,30,37)(15,18)(16,28)(17,33)(19,24,35,25)(27,31,34,32);(6,9)(7,14,15,30)(8,19)(10,25)(11,24)(12,23)(13,27,18,34)(16,32,17,31)(20,36)(21,35)(22,26)(28,37,33,29)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4298);