Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C5 · Schur multiplier (H2) = C214 |
| direct product | C5×#62652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C26 |
| commutator | C2 | C25×C4×C5 |
| Frattini, cosocle | C22 | C26×C5 |
| prime radical | C22×C5×H(5,𝔽2) | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
11674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 6 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,16)(13,20)(14,15)(17,21)(18,22)(19,25)(23,24);(11,15)(16,22)(17,20)(23,25);(12,18)(16,22)(19,24)(23,25);(10,12)(11,16)(13,19)(14,18)(15,22)(17,23)(20,25)(21,24);(10,11)(12,16)(13,17)(14,15)(18,22)(19,23)(20,21)(24,25);(10,13)(11,17)(12,19)(14,21)(15,20)(16,23)(18,24)(22,25)650 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55640);