Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C25 | C23×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,18)(13,16)(14,17)(15,20)(19,22)(21,23)(24,25);(10,12)(11,15,17,18)(13,19)(14,20)(16,22,23,24)(21,25);(6,8,9,7)(10,13)(11,16)(12,19)(14,21)(15,22)(17,23)(18,24)(20,25)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3903);