Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60811 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,28)(13,19)(14,18)(15,21)(16,25)(17,32)(20,30)(23,34)(24,35)(26,33)(27,36)(29,37)(31,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11,14,25)(10,16,18,22)(12,29,31,13)(15,26,30,27)(17,34,35,39)(19,38,37,28)(20,33,21,36)(23,32,40,24);(3,8)(5,7)(9,12,30,32,14,31,15,24)(10,17,20,38,18,35,21,28)(11,23,26,13,25,40,27,29)(16,19,36,39,22,37,33,34);(2,6)(5,7)(10,18)(16,22)(17,35)(19,37)(20,21)(28,38)(33,36)(34,39);(9,13,30,40,14,29,15,23)(10,19,20,39,18,37,21,34)(11,24,26,12,25,32,27,31)(16,17,36,38,22,35,33,28)Polycyclic presentation:
2^9|b^2=f,c^2=fh,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717595);