Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
868 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)(6,7)(8,9)(10,11)(12,15)(13,17)(14,16)(18,19)(20,24)(21,26)(22,25)(23,27)(28,33)(29,31)(30,32);(7,9)(10,12,14,13)(11,15,16,17)(18,20)(19,24)(21,29)(22,30)(23,28)(25,32)(26,31)(27,33);(10,13,14,12)(11,15,16,17)(19,25)(20,28)(21,23)(24,31)(29,30)(32,33);(7,9)(18,21,22,23)(19,26,25,27)(20,29,30,28)(24,31,32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19137);