Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#34977 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2962 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)(10,11)(12,16)(13,17)(14,15)(18,19)(20,23)(21,22)(24,25);(7,9)(18,19)(20,24)(21,22)(23,25);(11,15)(16,17)(19,22)(23,24);(10,12,14,13)(11,16,15,17)(18,20,21,25)(19,23,22,24);(10,13,14,12)(11,17,15,16)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27965);