Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,7,4)(10,13)(11,15)(17,18,22,19)(20,24,21,23)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,40,36,39);(1,3,5,7)(2,6,8,4)(10,13)(14,16)(17,19,22,18)(20,24,21,23)(26,30)(31,38)(32,35)(39,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,23,19,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,2)(3,6)(4,7)(5,8)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4,5,6)(2,7,8,3)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,24,19,23)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1194378);