Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11024 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C2×D4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,13)(14,15,16,17)(19,24)(21,22)(25,28)(26,27)(29,30)(32,33);(7,9)(18,19,23,27)(20,28,31,32)(21,26,22,24)(25,29,33,30);(11,13)(18,20,23,31)(19,25,27,33)(21,30,22,29)(24,28,26,32)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4012);