Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10418 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
488 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)(7,8)(10,11)(12,15)(13,16)(14,17)(18,19,22,20)(21,23,25,24);(6,7)(8,9)(10,12,14,13)(11,15,17,16)(18,20,22,19)(21,24,25,23);(6,8,9,7)(11,16)(12,13)(15,17)(19,23)(20,24)(21,25)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3406);