Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 247 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 — 3 generators on 16 points:
(1,2,3,4);(5,6,7,8)(9,10)(11,13)(12,15)(14,16);(10,13,15,14)112 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#339, #105367, #105368, … |
| maximal normal subgroup of | C8×#339, #105367, #105368, … |
| maximal quotient of | C4×#1276, #105267, #105269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3688);