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) = C23×C4 |
| direct product | C5×#22808 |
| 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 | |
782 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)(6,7,8,9)(10,11)(12,15)(13,17)(14,16)(18,19,22,25,23,26,30,21)(20,27,28,32,29,33,31,24);(10,12,14,13)(11,15,16,17)(18,20,23,29)(19,24,26,32)(21,27,25,33)(22,31,30,28);(10,13,14,12)(11,15,16,17);(18,21,30,26,23,25,22,19)(20,24,31,33,29,32,28,27)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15796);