Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1936 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,7,12,9,13,8,23)(3,5,11,24,16,14,26,6)(4,17,21,32,22,18,27,15)(10,28,29,20,31,19,30,25);(1,3,14,12,8,11,24,13,9,16,5,23,7,26,6,2)(4,18,19,10,21,32,25,30,22,17,28,31,27,15,20,29);(1,4,5,19,8,21,6,25,9,22,14,28,7,27,24,20)(2,10,3,15,12,30,11,18,13,31,16,32,23,29,26,17)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=d,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13158, #494936, #495899, … |
| maximal normal subgroup of | C2×#13158, #494936, #495899, … |
| maximal quotient of | #140190, #140212, #140283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6146);