Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C8×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | 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×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
455 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 223 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);(3,4,5,6,7,8,9,10);(11,12,13,14,15,16,17,18);(12,18)(13,17)(14,16)112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7068, #122442, #122450, … |
| maximal normal subgroup of | C2×#7068, #122442, #122450, … |
| maximal quotient of | C2×C8×#102, C2×C8×#107, C4×C8×D8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10830);