Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
223 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,7)(3,5,8,6)(13,16)(18,21)(19,22)(23,24);(2,5)(3,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,21)(19,22,23,24);(9,11,12,15)(10,13,14,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=gh,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=eg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7776, #165741, #165761, … |
| maximal normal subgroup of | C2×#7776, #165741, #165761, … |
| maximal quotient of | #97541, #99370, #99377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 764);