Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#29671 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1132 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)(12,15)(13,14)(16,17)(18,19,22,26)(20,23,27,31)(21,24,29,25)(28,30,33,32);(6,7)(8,9)(11,14)(15,17)(18,20,22,27)(19,23,26,31)(21,28,29,33)(24,30,25,32);(7,9)(12,16)(15,17)(19,24)(21,29)(23,30)(25,26)(28,33)(31,32);(10,12,13,16)(11,15,14,17)(20,27)(23,31)(28,33)(30,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22659);