Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#8276 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C43×C5 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11,13,15,14,16,12,17);(11,15)(12,13)(16,17)(18,19,20,23,21,22,24,25);(6,8,9,7)(11,16)(15,17)(19,22)(23,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1264);