Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | Aut(G) = C24×#3109 (order 2048) · Inn(G) = #442 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,6)(4,5)(7,8)(9,10,11,15,14,19,13,17)(12,21,20,24,22,16,23,18);(2,5)(6,8)(10,15,19,17)(12,22)(16,24,21,18)(20,23);(1,3,4,7)(2,6,5,8)(10,16)(11,13)(12,20)(15,24)(17,18)(19,21)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=dg,d^2=h,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,d^c=dh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13119, #494815, #494816, … |
| maximal normal subgroup of | C2×#13119, #494815, #494816, … |
| maximal quotient of | #140083, #140084, #140099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6107);