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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,14,21)(11,23,26,17)(12,27,24,19)(13,20,30,37)(15,18,16,22)(25,39,40,33)(28,36,29,35)(31,34,32,38);(2,6)(7,8)(9,11)(10,17)(12,15)(13,29)(14,26)(16,24)(18,19)(20,36)(21,23)(22,27)(25,31)(28,30)(32,40)(33,34)(35,37)(38,39);(10,18)(11,24)(12,26)(15,16)(19,27)(20,34)(21,22)(25,29)(28,40)(31,32)(35,36)(37,38);(1,3,5,4)(2,7,6,8)(9,12,14,24)(10,19,21,27)(11,15,26,16)(13,25,30,40)(17,18,23,22)(20,33,37,39)(28,32,29,31)(34,35,38,36);(1,4,5,3)(2,8,6,7)(9,13,14,30)(10,20,21,37)(11,25,26,40)(12,28,24,29)(15,31,16,32)(17,33,23,39)(18,34,22,38)(19,35,27,36)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=f,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, 6485477);