Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#10397 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(7,8)(10,11)(12,17)(13,16)(14,15)(18,19)(20,22)(21,23)(24,25);(6,7)(8,9)(10,12,13,14)(11,15,16,17)(18,20)(19,22)(21,24)(23,25);(6,8,9,7)(11,15)(12,14)(16,17)(22,25)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3385);