Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#35341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(7,9)(10,11,14,12)(13,16,17,15)(21,28)(25,30)(27,32)(31,33);(6,7,8,9)(10,12,14,11)(13,16,17,15)(19,23)(20,26)(25,30)(27,32);(10,13,14,17)(11,15,12,16)(18,19,22,23)(20,24,26,29)(21,25,28,30)(27,31,32,33);(18,20)(19,24)(21,27)(22,26)(23,29)(25,31)(28,32)(30,33);(18,21,22,28)(19,25,23,30)(20,27,26,32)(24,31,29,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28329);