Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10440 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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)(10,15)(11,18)(12,13)(16,19)(17,22)(20,21);(9,10,14,17)(11,19,21,16)(12,22,13,15)(18,24,20,23);(2,5)(3,8)(6,7)(9,11,12,20,14,21,13,18)(10,16,15,23,17,19,22,24)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,e^c=egh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10454, #239904, #239918, … |
| maximal normal subgroup of | C2×#10454, #239904, #239918, … |
| maximal quotient of | #104959, #104961, #104970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3442);