Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#1580 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C4×D4 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
447 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,8)(5,7,10,9)(11,12,14,17,15,13,16,18);(4,7)(5,10)(8,9);(4,8)(7,9)(11,13,14,18,15,12,16,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | #128688, #128692, #128719, … |
| maximal normal subgroup of | #128688, #128692, #128719, … |
| maximal quotient of | C2×#8200, C2×#8284, C4×#1580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10836);