Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1592 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)(3,7)(4,6)(5,8)(10,14)(12,15)(19,23)(22,24)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(1,3,6,5)(2,7,4,8)(10,14)(11,16)(18,21)(22,24)(26,27)(29,37)(31,33)(35,40);(1,2,6,4)(3,8,5,7)(9,10,13,14)(11,15,16,12)(17,18,20,21)(19,22,23,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,14)(13,16)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,11,14,16)(17,19,20,23)(18,22,21,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419645);