Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | C2×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
529 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(2,5)(3,7)(6,8)(9,10,11,13,12,14,15,16);(1,2)(3,8)(4,6)(5,7);(3,7)(5,8)Polycyclic presentation:
2^8|a^2=f,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93677, #136817, #136818, … |
| maximal normal subgroup of | #93677, #136817, #136818, … |
| maximal quotient of | #96938, #96940, #96943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1306);