Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| 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 | #3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,13,19,15,20,14,17)(11,16,23,31,24,29,21,32)(12,25,26,33,27,34,28,18)(22,36,35,39,38,40,37,30);(1,3,5,7)(2,6,8,4)(9,11,15,24)(10,16,20,29)(12,22,27,38)(13,23,14,21)(17,32,19,31)(18,30,33,39)(25,36,34,40)(26,35,28,37);(1,4,5,6)(2,7,8,3)(10,17)(11,21)(13,14)(16,29)(18,25)(19,20)(22,35)(23,24)(26,28)(30,39)(33,34)(37,38);(2,8)(3,7)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,30)(17,25)(19,34)(20,33)(21,35)(23,37)(24,38)(29,39)(31,40)(32,36)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6669046, #6669047, #6669048, … |
| maximal normal subgroup of | #6669046, #6669047, #6669048, … |
| maximal quotient of | #344977, #344979, #344982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24217);