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 | C2×C4×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| socle | C23 | C25 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
5605 conjugacy classes of subgroups; only the 5295 normal subgroups are shown.
This group has 5605 conjugacy classes of subgroups — too many to draw. Only its 5295 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4,5,6);(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,20)(19,22);(7,9,11,17)(8,12,14,15)(10,18,19,16)(13,21,22,20);(7,10,11,19)(8,13,14,22)(9,16,17,18)(12,20,15,21);(8,14)(12,15)(13,22)(20,21)136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55638);