Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,4)(5,7)(10,14)(12,16)(17,18,20,22)(19,23,24,21)(25,26,30,31)(27,33,34,28)(29,37,38,32)(35,36,40,39);(1,2,6,8)(3,7,4,5)(19,24)(21,23)(26,31)(27,34)(29,38)(36,39);(1,3,6,4)(2,7,8,5)(9,10,13,14)(11,12,15,16)(17,19)(18,21)(20,24)(22,23)(25,27,30,34)(26,28,31,33)(29,35,38,40)(32,36,37,39);(1,4,6,3)(2,5,8,7)(9,11,13,15)(10,12,14,16)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,5)(2,3)(4,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1027621);