Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,18)(16,19)(17,20);(2,4)(5,6)(7,11)(8,10)(9,12)(14,18)(15,16)(19,20);(6,10)(7,8)(11,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^8|d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dg,e^c=eh,f^c=fgh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #10304⋊C3 |
| maximal subgroup of | #94642, #125647, #125721, … |
| maximal normal subgroup of | #94642, #125647, #125721, … |
| maximal quotient of | #102785, #102795, #102857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3292);