Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14)(11,17)(12,19)(15,22)(16,23)(18,21)(20,24);(2,5)(3,7)(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,23)(22,24);(10,15)(14,22)(16,21)(18,23)Polycyclic presentation:
2^8|a^2=e,d^2=gh,e^2=fgh,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #128923, #128944, #129994, … |
| maximal normal subgroup of | #128923, #128944, #129994, … |
| maximal quotient of | #122879, #122882, #122883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4918);