Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C214 |
| direct product | C23×C5×#123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C5 | C23 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| prime radical | C24×C5×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C5 ◁ G | |
15150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 6 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4);(5,6);(8,10)(12,15)(13,17);(11,12)(13,16)(14,15)(17,18);(7,8)(9,10)(11,13,14,17)(12,16,15,18)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55646);