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) = C23×C4 |
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 | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,16)(12,27)(13,30)(14,20)(15,21)(17,29)(18,31)(19,32)(22,37)(23,39)(24,38)(25,35)(26,36)(28,40)(33,34);(2,5)(3,8)(9,11)(10,16)(12,28)(13,23)(14,24)(15,26)(17,37)(18,34)(19,35)(20,38)(21,36)(22,29)(25,32)(27,40)(30,39)(31,33);(9,12,15,18)(10,17,21,13)(11,22,26,39)(14,30,32,29)(16,33,36,40)(19,31,20,27)(23,38,37,35)(24,28,25,34);(1,3,4,8)(2,6,5,7)(9,13,15,17)(10,18,21,12)(11,23,26,37)(14,27,32,31)(16,34,36,28)(19,29,20,30)(22,38,39,35)(24,40,25,33)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^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 | #6670992, #6670999, #6671004, … |
| maximal normal subgroup of | #6670992, #6670999, #6671004, … |
| maximal quotient of | #340432, #340433, #340436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23929);