Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7092 |
| automorphism | Aut(G) = #72280184 (order 4096) · Inn(G) = #331 · Out(G) = #339 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #331 |
| commutator | C4×C8 | C2×C4 |
| Frattini, cosocle | C82 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7,8,15)(3,11)(4,5)(9,12,10,16)(13,14)(17,18,20,24,30,29,32,27,21,25,31,23,22,19,26,28);(1,2,6,8)(3,12,14,16)(4,10,5,9)(7,11,15,13)(17,19)(18,22)(20,24)(21,29)(23,31)(25,30)(26,27)(28,32)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=fh,d^2=ef,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7093, #122856, #122858, … |
| maximal normal subgroup of | C2×#7093, #122856, #122858, … |
| maximal quotient of | #92867, #93065, C2×#7093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 81);