Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25861 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
1282 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)(10,11)(12,16)(13,15)(14,17)(18,19,22,25)(20,26,27,33)(21,29,30,24)(23,31,32,28);(6,7)(8,9)(10,12,14,13)(11,15,17,16);(7,9)(10,12,14,13)(11,16,17,15)(18,20,21,28,22,27,30,31)(19,23,24,26,25,32,29,33);(10,13,14,12)(11,16,17,15)(20,27)(23,32)(26,33)(28,31)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18849);