Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#1293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #95 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C25×C5 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C24×C4×C5 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4)(5,9,10,7)(6,8)(11,12,14,19,16,20,15,18)(13,21,22,17,24,25,23,26);(3,5)(4,7)(6,10)(8,9)(11,13,15,23,16,24,14,22)(12,17,19,25,20,26,18,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 993);