Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C26 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | #121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
218 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,17,15,11,13,19)(12,16,21,23,22,18,20,24);(2,5,7,6)(9,11,15,10)(12,16,22,18)(13,17,14,19)(20,24,21,23);(9,12,14,21,15,22,13,20)(10,16,17,23,11,18,19,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128654, #128657, C2×#8249, … |
| maximal normal subgroup of | #128654, #128657, C2×#8249, … |
| maximal quotient of | #96778, #96779, #96782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1237);