Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20943 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,14,19,15,17,13,18)(12,20,22,16,23,25,21,24);(6,7,8,9)(10,12)(11,16)(13,22)(14,21)(15,23)(17,24)(18,25)(19,20);(10,12)(11,16)(13,21)(14,22)(15,23)(17,24)(18,20)(19,25);(11,17)(16,24)(18,19)(20,25)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13931);