Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C4×C5×#563 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4)(29,30,31,32,33);(6,9)(7,11)(13,14,18,16)(15,21,23,19)(17,24,26,20)(22,27,25,28);(6,9)(10,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,23,21)(17,24,26,20)(22,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27342);