Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C23×C4×D4 |
| unambiguous semidirect product | C4⋊(C24×C4) |
| unambiguous semidirect product | C22⋊(C24×C4) |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C22 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| socle | C25 | C23 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
9999 conjugacy classes of subgroups; only the 5839 normal subgroups are shown.
This group has 9999 conjugacy classes of subgroups — too many to draw. Only its 5839 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 14 points:
(1,2);(3,4);(5,6);(7,8,9,10);(11,12,13,14);(12,14)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55630);