Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#34129 |
| 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×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(11,15)(16,17)(18,19)(20,23)(21,25)(22,24);(10,11,14,15)(12,16,13,17)(18,20,23,19)(21,25,22,24);(10,12,14,13)(11,16,15,17)(18,21,23,22)(19,24,20,25);(10,12,14,13)(11,17,15,16)(18,22,23,21)(19,24,20,25);(10,13,14,12)(11,17,15,16)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27117);