Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1993 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1993 |
| 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 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,14,18,30)(4,20,23,28)(5,24)(6,11)(7,13)(8,9,22,27)(10,15)(12,26)(16,25)(17,21,19,32)(29,31);(1,3,15,28,26,17,5,9,7,18,31,20,6,19,25,27)(2,8,16,32,11,23,29,30,13,22,24,21,12,4,10,14);(1,4,7,23)(2,9)(3,16)(5,21,25,32)(6,22,26,8)(10,19)(11,20)(12,28)(13,27)(14,15,30,31)(17,29)(18,24)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=cg,e^b=eh,f^b=fh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13291, #495392, #495573, … |
| maximal normal subgroup of | C2×#13291, #495392, #495573, … |
| maximal quotient of | #141905, #141906, #141926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6279);