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) = C2 |
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 | |
153 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 153 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:
(1,2,5,8,4,7,3,6)(9,10,14,16,13,17,12,21)(11,18,22,23,19,24,20,15);(9,11,12,20,13,19,14,22)(10,15,16,18,17,23,21,24);(1,2,4,7)(3,8,5,6)(11,19)(12,14)(16,21)(18,24)Polycyclic presentation:
2^8|a^2=deh,b^2=d,c^2=h,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×#10609, #242676, #242677, … |
| maximal normal subgroup of | C2×#10609, #242676, #242677, … |
| maximal quotient of | #105192, #105195, #105196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3597);