Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C5×#1969 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C2×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
1614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(27,28,29,30,31);(4,6)(7,8,9,10)(11,12)(13,18)(14,23)(15,25)(16,20)(17,21)(19,24)(22,26);(3,4,5,6)(8,10)(11,13,15,21,16,22,24,14)(12,17,19,18,20,23,25,26);(11,14,24,22,16,21,15,13)(12,18,25,17,20,26,19,23)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14210);