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×#35312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2262 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)(7,8)(12,16)(14,17)(18,19,23,20)(21,24,28,26)(22,27,30,25)(29,32,33,31);(6,7,9,8)(10,11,13,15)(12,14,16,17)(19,20)(21,28)(25,27)(29,33);(6,8,9,7)(10,12)(11,14)(13,16)(15,17)(18,20,23,19)(21,26,28,24)(22,27,30,25)(29,32,33,31);(18,21)(19,24)(20,26)(22,29)(23,28)(25,31)(27,32)(30,33);(18,22,23,30)(19,25,20,27)(21,29,28,33)(24,31,26,32)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28300);