Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D8×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
345 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4,5,6,7,8);(2,8)(3,7)(4,6);(9,10,11,12,13,14,15,16)(17,24,23,22,21,20,19,18);(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)49 conjugacy classes — view the full character table.
| maximal subgroup of | #123283, #123293, #123305, … |
| maximal normal subgroup of | #123283, #123293, #123305, … |
| maximal quotient of | Q16×#102, D8×#103, D8×#107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12962);