Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#3168 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C23×C5 | C22×D4 |
| prime radical | C2×C5×#466 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
2070 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)(5,8)(6,7)(9,10)(11,12)(13,17)(14,21)(15,24)(16,19)(18,22)(20,25)(23,26);(4,7)(8,10)(11,13,16,15)(12,17,19,24)(14,20,23,22)(18,21,25,26);(11,14,15,22,16,23,13,20)(12,18,17,21,19,25,24,26);(3,5,6,9)(4,8,7,10)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53658);