Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C22×#518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
7295 conjugacy classes of subgroups; only the 5327 normal subgroups are shown.
This group has 7295 conjugacy classes of subgroups — too many to draw. Only its 5327 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 16 points:
(3,4);(1,2)(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)(10,14)(12,16)(13,15)136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55633);