Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C23×D4×#392 (order 4096) · Inn(G) = D4 · Out(G) = C23×#392 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C8×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
277 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 277 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) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,5,4,6,7,8);(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16)112 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#104, #128641, #128642, … |
| maximal normal subgroup of | C16×#104, #128641, #128642, … |
| maximal quotient of | C8×D4⋊C8, C8×Q8⋊C8, #96692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1178);