Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C2×#335 |
| 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:
(1,2,5,10,6,12,4,20,7,13,15,9,21,11,22,27)(3,14,18,25,19,29,17,8,16,24,30,26,32,28,31,23);(1,3,15,30,6,19,22,31,7,16,5,18,21,32,4,17)(2,8,20,28,11,25,10,24,13,23,27,29,12,26,9,14);(3,16)(8,23)(14,24)(17,31)(18,30)(19,32)(25,26)(28,29)Polycyclic presentation:
2^8|a^2=e,b^2=df,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #94407, #94408, #129121, … |
| maximal normal subgroup of | #94407, #94408, #129121, … |
| maximal quotient of | #122931, #122935, #122936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4999);