Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | Aut(G) = #72315303 (order 2048) · Inn(G) = #390 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1955 |
| prime radical | #525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,6,12,7,11,26,30,8,13,27,16,5,10,15,21)(3,14,18,32,17,24,29,31,19,22,9,28,25,4,20,23);(1,3,15,29,5,25,27,18,8,19,26,20,7,17,6,9)(2,4,21,32,10,22,16,23,13,24,30,28,11,14,12,31);(1,4,5,22,8,24,7,14)(2,9,11,20,13,18,10,29)(3,16,17,21,19,12,25,30)(6,23,15,28,27,31,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132072, #132073, C2×#13201, … |
| maximal normal subgroup of | #132072, #132073, C2×#13201, … |
| maximal quotient of | #140563, #140567, #140585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6189);