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×#35016 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(8,12)(11,13)(14,15)(16,20)(17,18)(19,21)(22,23,27,26)(24,29,28,25);(7,10)(11,13)(15,18)(19,20)(22,24)(23,25)(26,29)(27,28);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21)(22,25,27,29)(23,28,26,24);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21);(6,8)(7,11)(9,12)(10,13)(14,16,17,21)(15,19,18,20)(22,26,27,23)(24,29,28,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28004);