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) = C2×C4 |
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 | |
242 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)(11,14)(12,15)(13,16)(17,18)(19,21)(20,23)(22,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(19,24)(21,22);(10,14)(11,12)(15,16)(18,21)(19,24)(22,23)Polycyclic presentation:
2^8|a^2=d,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 | #10358⋊C3 |
| maximal subgroup of | #151381, #151398, #151476, … |
| maximal normal subgroup of | #151381, #151398, #151476, … |
| maximal quotient of | #104709, #104712, #104719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3346);