Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
269 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 71 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)(10,14,16,19)(11,17,18,20);(9,10,12,19,13,16,21,14)(11,15,20,23,18,24,17,22);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,18)(14,22)(16,24)(17,21)(19,23)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=g,e^2=gh,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9053, #195474, #195475, … |
| maximal normal subgroup of | C2×#9053, #195474, #195475, … |
| maximal quotient of | #99595, #99596, #99599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2041);