Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11351 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #381 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C25 | C23×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
850 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)(10,11)(12,15,19,23)(13,20)(14,17)(16,21)(18,22,24,25);(6,7)(8,9)(10,12)(11,15,17,23)(13,18)(14,19)(16,22,20,25)(21,24);(7,9)(10,13)(11,16,17,20)(12,18)(14,21)(15,22,23,25)(19,24)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4339);