Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C42×SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 327 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 327 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) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6,7,8);(9,10,11,12,13,14,15,16);(10,12)(11,15)(14,16)112 conjugacy classes — view the full character table.
| Fitting subgroup of | SD16×C42⋊C3 |
| maximal subgroup of | C4×C8×SD16, C4×SD16⋊C8, #183878, … |
| maximal normal subgroup of | C4×C8×SD16, C4×SD16⋊C8, #183878, … |
| maximal quotient of | C4×C8×SD16, #183884, #183908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13486);