Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,12,15,14)(17,18)(19,21)(20,22)(23,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18,20,22)(19,23,24,21)(28,36)(31,38)(34,40)(37,39);(3,8)(5,7)(11,15)(12,14)(17,18)(19,23)(20,22)(21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,12,13,14)(10,15,16,11)(17,19,20,24)(18,21,22,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393483);