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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | 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 | |
293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,7)(8,11,9,12)(14,18)(15,19)(16,20);(2,4)(6,11)(7,12)(8,9)(13,14,17,15)(16,19,20,18);(5,7,10,6)(8,12,9,11)(13,15,17,14)(16,18,20,19)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,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 | #10316⋊C3 |
| maximal subgroup of | #94657, #125895, #125950, … |
| maximal normal subgroup of | #94657, #125895, #125950, … |
| maximal quotient of | #102830, #102836, #102869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3304);