Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = D8 · Out(G) = C25×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
| direct product | C8×SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
121 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 121 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 24 points:
(1,2,3,4,5,6,7,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(10,16)(11,23)(12,14)(13,21)(15,19)(18,24)(20,22)88 conjugacy classes — view the full character table.
| maximal subgroup of | C16×SD32, #122944, #122948, … |
| maximal normal subgroup of | C16×SD32, #122944, #122948, … |
| maximal quotient of | C16×SD32, #122948, #122954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5029);