Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22897 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#453 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,21)(13,17)(14,19)(15,18)(16,20)(22,25)(23,24);(6,8,9,7)(10,12,16,22)(11,13,20,23)(14,24,15,17)(18,25,19,21);(10,13,16,23)(11,12,20,22)(14,21,15,25)(17,19,24,18);(10,13)(11,17)(12,18)(14,25)(15,21)(16,23)(19,22)(20,24)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15885);