Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1668 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)(3,6,4,7)(9,10,13,15)(11,16,14,12)(25,26,30,27)(28,31,35,33)(29,32,38,34)(36,39,37,40);(1,3,5,4)(2,6,8,7)(11,14)(12,16)(18,21)(22,24)(26,27)(28,35)(32,34)(36,37);(1,3,5,4)(2,7,8,6)(9,11,13,14)(10,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,26)(28,33,35,31)(29,32,38,34)(36,39,37,40);(1,4,5,3)(2,6,8,7)(17,19,20,23)(18,22,21,24)(25,28)(26,31)(27,33)(29,37)(30,35)(32,40)(34,39)(36,38);(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,15)(13,16)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fg,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3961174);