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 | #537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,25)(13,17)(14,19)(15,20)(18,26)(21,30)(22,36)(23,31)(24,32)(27,33)(28,34)(29,35)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,24)(10,16,20,32)(12,22,27,38)(13,23,14,21)(17,31,19,30)(18,29,34,39)(25,36,33,40)(26,35,28,37);(1,3,4,7)(2,5,6,8)(10,17)(11,21)(13,14)(18,25)(19,20)(22,35)(23,24)(26,28)(30,31)(33,34)(36,40)(37,38);(2,6)(5,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,29)(17,25)(19,33)(20,34)(21,35)(23,37)(24,38)(30,40)(31,36)(32,39)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=e,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 | #6669699, #6669702, #6669712, … |
| maximal normal subgroup of | #6669699, #6669702, #6669712, … |
| maximal quotient of | #336647, #336650, #336659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23685);