Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×D4×#525 (order 2048) · Inn(G) = D4 · Out(G) = C22×#525 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(1,2,5,8,6,3,4,7)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,3)(2,6)(4,8)(5,7);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,d^b=df112 conjugacy classes — view the full character table.
| maximal subgroup of | #128620, #128621, #128622, … |
| maximal normal subgroup of | #128620, #128621, #128622, … |
| maximal quotient of | #124099, #124130, #124134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4601);