Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×#391 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3087 conjugacy classes of subgroups; only the 1375 normal subgroups are shown.
This group has 3087 conjugacy classes of subgroups — too many to draw. Only its 1375 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(3,4);(1,2)(3,4);(6,8)(9,10,12,11)(15,19)(17,20);(5,6,7,8)(9,11,12,10)(13,14,16,18)(15,17,19,20);(13,15,16,19)(14,17,18,20)112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27009);