Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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×#577 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1865 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)(12,26)(13,20)(14,18)(15,21)(16,22)(19,24)(23,32)(25,33)(27,36)(28,40)(29,34)(30,38)(31,37)(35,39);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,22,33)(12,14,24,15)(13,27,30,39)(18,19,21,26)(20,36,38,35)(23,31,28,29)(32,37,40,34);(10,18)(11,12)(14,15)(19,26)(20,34)(21,22)(23,27)(24,25)(28,39)(29,31)(35,36)(37,38);(9,12)(10,19)(11,14)(13,28)(15,25)(16,24)(17,21)(18,33)(20,32)(22,26)(23,30)(27,29)(31,39)(34,35)(36,37)(38,40);(9,13)(10,20)(11,23)(12,27)(14,29)(15,31)(16,30)(17,32)(18,34)(19,35)(21,37)(22,38)(24,39)(25,28)(26,36)(33,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481782);