Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| 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 | |
321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,4,7,6)(9,10,13,17)(11,19,20,16)(12,21,22,24)(14,23,15,18);(1,3,5,7)(2,6,8,4)(9,11,13,20)(10,16,17,19)(12,18,22,23)(14,21,15,24);(1,4,5,6)(2,7,8,3)(9,12)(10,11)(13,22)(14,19)(15,16)(17,20)(18,24)(21,23);(2,8)(3,7)(9,11)(10,12)(13,20)(14,24)(15,21)(16,23)(17,22)(18,19)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6669694, #6669702, #6669708, … |
| maximal normal subgroup of | #6669694, #6669702, #6669708, … |
| maximal quotient of | #338683, #338686, #338687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23827);