Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C2×C5×#3510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| prime radical | C2×C5×#535 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 31 points:
(1,2)(27,28,29,30,31);(5,9)(7,10)(11,12)(13,17)(14,21)(15,16)(18,22)(19,25)(20,23)(24,26);(3,4,6,8)(5,7,9,10)(12,16)(17,23)(18,21)(24,25);(3,5,6,9)(4,7,8,10)(11,13,15,20)(12,17,16,23)(14,19,22,26)(18,24,21,25);(11,12)(13,17)(14,18)(15,16)(19,24)(20,23)(21,22)(25,26);(11,14,15,22)(12,18,16,21)(13,19,20,26)(17,24,23,25)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55796);