Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | Aut(G) = C24×#442 (order 1024) · Inn(G) = #442 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,5,3,4,6,8,7)(9,10,11,16,13,18,19,17)(12,21,20,15,23,14,22,24);(2,6)(3,7)(10,14)(11,19)(12,22)(15,16)(17,24)(18,21)(20,23);(1,3,5,6,4,7,8,2)(9,11,13,19)(12,22,23,20)(14,21)(15,24)Polycyclic presentation:
2^8|a^2=e,c^2=ef,d^2=f,e^2=g,f^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fh,c^b=cg,d^b=dfh,e^b=eh,f^b=fh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12087, #483836, #483837, … |
| maximal normal subgroup of | C2×#12087, #483836, #483837, … |
| maximal quotient of | #128738, #128739, #129177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5075);