Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,22)(11,23,26,32)(12,18,31,37)(13,19,17,25)(14,20,15,21)(24,38,39,34)(27,33,40,28)(29,35,30,36);(9,11,16,26)(10,17,22,13)(12,27,31,40)(14,25,15,19)(18,34,37,38)(20,23,21,32)(24,29,39,30)(28,36,33,35);(1,3,5,4)(2,6,7,8)(9,12)(10,18)(11,24)(13,28)(14,30)(15,29)(16,31)(17,33)(19,27)(20,36)(21,35)(22,37)(23,38)(25,40)(26,39)(32,34);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,20)(12,28)(14,32)(15,23)(16,17)(18,27)(21,26)(22,25)(24,36)(29,38)(30,34)(31,33)(35,39)(37,40);(9,14,16,15)(10,20,22,21)(11,25,26,19)(12,29,31,30)(13,32,17,23)(18,35,37,36)(24,27,39,40)(28,38,33,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358968);