Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #121 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #388 |
| prime radical | C23 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
173 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,4,7,5,3,6,8)(10,14)(11,17)(12,19)(15,22)(16,21)(20,23);(1,3,4,8,5,2,6,7)(9,10,12,14,13,16,19,21)(11,15,18,20,17,23,24,22);(9,11,12,18,13,17,19,24)(10,15,14,20,16,23,21,22)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^c=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9549, #202478, #202479, … |
| maximal normal subgroup of | C2×#9549, #202478, #202479, … |
| maximal quotient of | #101895, #101896, #101899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2537);