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×#3502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
6826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 6 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4,6,8)(5,9,10,7)(11,12,16,20)(13,18,14,17)(15,19,23,26)(21,25,22,24);(11,13,16,14)(12,17,20,18)(15,22,23,21)(19,25,26,24);(3,5,6,10)(4,7,8,9)(11,14,16,13)(12,18,20,17)(15,22,23,21)(19,25,26,24);(11,15,16,23)(12,19,20,26)(13,21,14,22)(17,24,18,25);(11,12)(13,17)(14,18)(15,19)(16,20)(21,24)(22,25)(23,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55772);