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×#35039 |
| 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 | |
1708 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)(14,15,19,21)(16,18,20,17)(22,23,26,25)(24,28,29,27);(7,10)(11,13)(14,16,19,20)(15,17,21,18)(22,24,26,29)(23,27,25,28);(6,7)(8,11)(9,10)(12,13)(14,15)(16,17)(18,20)(19,21);(6,7)(8,11)(9,10)(12,13)(14,17,19,18)(15,16,21,20);(6,8)(7,11)(9,12)(10,13)(14,18,19,17)(15,20,21,16)(22,25,26,23)(24,28,29,27)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28027);