Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#34993 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#534 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5)(7,8)(12,16)(15,17)(20,24)(21,25);(6,7,9,8)(11,14)(15,17)(18,19,22,23)(20,25,24,21);(6,8,9,7)(10,11)(12,15)(13,14)(16,17)(18,20)(19,21)(22,24)(23,25);(10,11)(12,15)(13,14)(16,17)(18,19)(20,21)(22,23)(24,25);(10,12)(11,15)(13,16)(14,17)(18,21)(19,20)(22,25)(23,24)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27981);