Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator, socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,14,12,13,15,16);(2,5)(6,8)(17,18,19,22,20,21,23,24);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,21)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,e^2=g,f^2=h,b^a=bd,c^a=cg,c^b=ch88 conjugacy classes — view the full character table.
| Fitting subgroup of | #8136⋊C3 |
| maximal subgroup of | #93618, #99293, #99294, … |
| maximal normal subgroup of | #93618, #99293, #99294, … |
| maximal quotient of | #96668, #96669, #105250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1124);