Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#10742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(11,16)(13,14)(17,24)(18,19)(20,22)(23,25);(10,11,14,19,15,18,13,16)(12,17,20,24,21,25,22,23);(10,12)(11,17)(13,22)(14,20)(15,21)(16,23)(18,25)(19,24)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3730);