Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | C2×#576 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,32)(24,31)(25,33)(26,35)(27,36)(28,34)(29,38)(30,37)(39,40);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,22,33)(12,15,23,14)(13,27,29,26)(18,19,21,32)(20,36,38,35)(24,30,39,28)(31,37,40,34);(10,18)(11,23)(12,25)(14,15)(19,32)(20,34)(21,22)(24,27)(26,39)(28,30)(35,36)(37,38);(9,12,16,23)(10,19,22,32)(11,14,25,15)(13,24,29,39)(17,18,33,21)(20,31,38,40)(26,30,27,28)(34,35,37,36);(9,13)(10,20)(11,24)(12,26)(14,28)(15,30)(16,29)(17,31)(18,34)(19,35)(21,37)(22,38)(23,27)(25,39)(32,36)(33,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481905);