Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #106 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C2×#334 |
| prime radical | C23 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,7,12,22,28,8,13,21,10,6,24,4,20)(3,14,17,26,19,29,16,25,15,27,23,32,18,31,30,9);(1,3,4,16,6,18,21,17,8,15,22,30,7,19,5,23)(2,9,10,14,12,26,20,29,13,25,11,27,24,32,28,31);(3,15)(9,25)(14,27)(16,30)(17,23)(18,19)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=d,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129121, #129125, #129137, … |
| maximal normal subgroup of | #129121, #129125, #129137, … |
| maximal quotient of | #122920, #122923, #122924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4990);