Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(10,11)(12,19)(13,17)(14,22)(15,18)(16,20)(21,24)(23,25);(6,7)(8,9)(13,14)(17,22)(19,24)(20,23);(7,9)(10,12)(11,16)(13,20)(14,23)(15,21)(17,24)(18,25)(19,22);(10,13)(11,17)(12,20)(14,15)(16,24)(18,22)(19,25)(21,23)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16900);