Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30867 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,16,20,15,22,14,21)(11,23,27,17,28,36,26,37)(12,29,30,18,32,25,31,38)(13,19,34,39,35,40,33,24);(1,2,5,6)(3,4,8,7)(9,11)(10,17)(12,24)(13,18)(14,26)(15,28)(16,27)(19,30)(20,36)(21,23)(22,37)(25,34)(29,33)(31,40)(32,39)(35,38);(1,3)(2,7)(4,6)(5,8)(9,12)(10,18)(11,24)(13,17)(14,30)(15,32)(16,31)(19,26)(20,29)(21,25)(22,38)(23,34)(27,40)(28,39)(33,36)(35,37);(2,6)(3,8)(9,13)(10,19)(11,25)(12,17)(14,33)(15,35)(16,34)(18,26)(20,39)(21,24)(22,40)(23,31)(27,38)(28,29)(30,36)(32,37);(1,4,5,7)(2,3,6,8)(9,14,15,16)(10,20,22,21)(11,26,28,27)(12,30,32,31)(13,33,35,34)(17,36,37,23)(18,29,38,25)(19,39,40,24)Polycyclic presentation:
2^9|a^2=gi,b^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089208);