Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#29817 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
866 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)(18,19,22,25,23,24,31,32)(20,21,27,28,26,29,33,30);(6,7)(8,9)(11,14)(15,17)(18,20)(19,21)(22,27)(23,26)(24,29)(25,28)(30,32)(31,33);(7,9)(12,16)(15,17)(18,21,23,29)(19,20,24,26)(22,30,31,28)(25,33,32,27);(10,12,13,16)(11,15,14,17)(19,24)(20,26)(25,32)(27,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22805);