Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#23063 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(11,15)(16,17)(18,19,22,25)(20,26,28,33)(21,29,31,24)(23,30,32,27);(6,8,9,7)(10,11)(12,16)(13,17)(14,15);(10,12,14,13)(11,16,15,17)(18,20,22,28)(19,23,25,32)(21,30,31,27)(24,33,29,26);(10,13,14,12)(11,16,15,17)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16051);