Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,6,4,8)(9,10,15,19)(11,21,17,23)(12,22,13,16)(14,20,24,18);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,19)(14,16)(15,17)(18,23)(20,21)(22,24);(1,4,5,3)(2,6,7,8)(9,12,15,13)(10,16,19,22)(11,20,17,18)(14,23,24,21);(9,13,15,12)(10,17,19,11)(14,23,24,21)(16,18,22,20)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=f,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26405, #6439251, #6439252, … |
| maximal normal subgroup of | C2×#26405, #6439251, #6439252, … |
| maximal quotient of | #307898, #307899, #307910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19393);