Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#25933 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#459 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×Q8 ◁ G | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,14,17)(12,16,13,15)(19,22)(20,23)(24,25);(10,12,14,13)(11,15,17,16)(19,22)(20,23)(24,25);(6,7,8,9)(10,12,14,13)(11,16,17,15)(18,19,23,25,21,24,20,22);(10,13,14,12)(11,16,17,15)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18921);