Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25937 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×Q8 ◁ G | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(7,9)(10,11,14,17)(12,16,13,15)(18,19,22,26)(20,29,30,25)(21,32,31,33)(23,28,24,27);(10,12,14,13)(11,15,17,16)(18,20,22,30)(19,25,26,29)(21,23,31,24)(27,33,28,32);(6,7,8,9)(10,12,14,13)(11,16,17,15)(18,21,24,20,22,31,23,30)(19,25,28,33,26,29,27,32);(10,13,14,12)(11,16,17,15)(19,26)(25,29)(27,28)(32,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18925);