Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #396 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
161 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,5,6,4,8,3,7)(9,10,15,18)(11,13,17,14)(12,19,22,23)(16,20,24,21);(9,11,13,18,15,17,14,10)(12,19,20,16,22,23,21,24);(2,6)(3,5)(7,8)(9,12,15,22)(10,16,18,24)(11,19,17,23)(13,21,14,20)Polycyclic presentation:
2^8|a^2=deh,b^2=d,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10608, #242670, #242678, … |
| maximal normal subgroup of | C2×#10608, #242670, #242678, … |
| maximal quotient of | #105190, #105191, #105194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3596);