Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#1335 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | D4⋊C8 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,7,12,9,13,18,11,10,14,6,17,8,5,15,16);(3,5)(4,9)(6,11)(7,16)(8,14)(10,13)(12,18)(15,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1335, #128526, #128570, … |
| maximal normal subgroup of | C4×#1335, #128526, #128570, … |
| maximal quotient of | C2×#7068, C2×#7069, #122440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4589);