Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C5 · Schur multiplier (H2) = C214 |
| direct product | C22×C5×#520 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C2 | C25×C4×C5 |
| Frattini, cosocle | C22 | C26×C5 |
| prime radical | C23×C5×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
12270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2)(17,18,19,20,21);(3,4);(5,6,7,8)(9,10)(11,15)(12,13)(14,16);(10,13)(14,15);(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55635);