Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#3166 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
1902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4,6,8)(5,9,10,7)(11,12,14,16)(13,19,20,26)(15,23,22,18)(17,25,24,21);(3,5,6,10)(4,7,8,9)(12,16)(17,24)(18,23)(19,26);(11,13,15,21,14,20,22,25)(12,17,18,19,16,24,23,26);(13,20)(17,24)(19,26)(21,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53650);