Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | Aut(G) = #72257321 (order 1024) · Inn(G) = #388 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #388 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1924 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
150 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 150 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,5,10,14,9,7,11,23,29,21,30,4,19)(3,13,15,28,17,8,20,26,18,27,22,31,16,25,32,24);(1,3,14,32,21,16,6,22,7,18,4,20,5,17,23,15)(2,8,9,28,30,13,12,24,11,25,19,31,10,27,29,26);(1,4,5,6,7,14,21,23)(2,9,10,29,11,19,30,12)(3,15,16,32,18,22,17,20)(8,24,13,31,25,26,27,28)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,d^b=dh,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13127, #494842, #495292, … |
| maximal normal subgroup of | C2×#13127, #494842, #495292, … |
| maximal quotient of | #140132, #140142, #140150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6115);