Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,15)(14,16)(17,18,20,23)(19,21,24,22);(1,3,4,8)(2,5,6,7)(9,11,12,10)(13,15,14,16)(19,24)(21,22);(10,15)(11,16)(13,14)(18,21)(19,24)(22,23)Polycyclic presentation:
2^8|a^2=dg,b^2=d,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #10360⋊C3 |
| maximal subgroup of | #94714, C2×#10360, #237891, … |
| maximal normal subgroup of | #94714, C2×#10360, #237891, … |
| maximal quotient of | #104713, #104718, #104725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3348);