Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 187 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)(3,4)(9,10,12,16,13,14,20,22)(11,17,18,24,19,15,21,23);(2,4)(5,6,7,8)(10,14)(12,20)(15,17)(18,21);(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93479, #93485, #93522, … |
| maximal normal subgroup of | #93479, #93485, #93522, … |
| maximal quotient of | #105569, #105571, #105577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3751);