Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7366 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1837 | C22 |
| prime radical | C23 | #355 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1837 ◁ G | |
| LECS | G ▷ #1837 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,6,11,29,42,7,13,28,43,22,12,30,44)(3,14,16,45,17,47,53,64,19,37,52,62,18,46,26,57)(4,20,23,41,24,40,27,58,25,39,56,63,54,9,36,38)(8,31,32,50,34,51,60,15,35,48,61,49,33,55,59,21);(1,3)(2,8,11,33,13,35,12,34)(4,21)(5,26,28,53)(6,18)(7,19)(9,37,20,47,40,14,39,46)(10,32,42,59,43,61,44,60)(15,25)(16,29,52,30)(17,22)(23,55,56,51)(24,49)(27,48,36,31)(38,62,41,64,58,45,63,57)(50,54)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=gi,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=degh,f^b=fi,d^c=dghi,e^d=ehi,g^d=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1779);