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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
357 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)(9,10,13,17)(11,18,20,24)(12,14,22,16)(15,21,23,19);(1,3,5,4)(2,6,7,8)(10,14)(12,22)(15,18)(16,17)(19,21)(23,24);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,22)(18,19)(20,23)(21,24);(9,11)(10,15)(12,21)(13,20)(14,18)(16,24)(17,23)(19,22)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=fh,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×#26406, #6439251, #6439253, … |
| maximal normal subgroup of | C2×#26406, #6439251, #6439253, … |
| maximal quotient of | #307901, #307904, #307905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19394);