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 |
| direct product | C5×#35012 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(8,12)(11,13)(14,15,19,17)(16,18,21,20)(23,26)(27,29);(7,10)(11,13)(15,17)(16,21)(22,23)(24,27)(25,26)(28,29);(6,7)(8,11)(9,10)(12,13)(14,16,19,21)(15,20,17,18);(6,7)(8,11)(9,10)(12,13)(14,17,19,15)(16,18,21,20)(22,24,25,28)(23,27,26,29);(6,8)(7,11)(9,12)(10,13)(14,18)(15,16)(17,21)(19,20)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28000);