Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19,13,12,14,18)(11,20,24,21,22,16,23,17);(1,3,5,2)(4,7,8,6)(9,11,13,22)(10,16,12,20)(14,23,15,24)(17,18,21,19);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,21,22,17)(14,18,15,19)(16,24,20,23);(2,3)(6,7)(9,11)(10,17)(12,21)(13,22)(14,24)(15,23)(16,19)(18,20)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6669698, #6669700, #6669702, … |
| maximal normal subgroup of | #6669698, #6669700, #6669702, … |
| maximal quotient of | #344016, #344019, #344020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24170);