Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11267 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C2×D4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
886 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)(10,11,12,13)(14,15,17,16)(19,24)(20,25)(21,22);(6,7)(8,9)(14,16,17,15)(18,19,22,20,23,25,21,24);(7,9)(15,16)(18,20,21,19,23,24,22,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4255);