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×C4 |
| direct product | C5×#25894 |
| 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 | |
1260 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)(12,16)(13,15)(14,17)(18,19,23,28)(20,26,31,25)(21,32,30,33)(22,29,24,27);(10,12,14,13)(11,15,17,16)(18,20)(19,25)(21,24)(22,30)(23,31)(26,28)(27,33)(29,32);(6,7,8,9)(10,12,14,13)(11,16,17,15)(18,21,22,31,23,30,24,20)(19,26,27,33,28,25,29,32);(10,13,14,12)(11,16,17,15)(18,22,23,24)(19,27,28,29)(20,30,31,21)(25,33,26,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18882);