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 | #2006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
372 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)(3,6)(4,5)(7,8)(9,10,14,11)(12,19,21,24)(13,15,23,17)(16,22,18,20);(10,15)(11,17)(13,23)(16,19)(18,24)(20,22);(2,5)(6,8)(9,11)(10,14)(12,18)(13,15)(16,21)(17,23)(19,22)(20,24);(1,3,4,7)(2,6,5,8)(9,12,14,21)(10,16,11,18)(13,22,23,20)(15,19,17,24)Polycyclic presentation:
2^8|a^2=h,d^2=f,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×#26580, #6439372, #6439373, … |
| maximal normal subgroup of | C2×#26580, #6439372, #6439373, … |
| maximal quotient of | #309452, #309453, #309454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19568);