Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94560 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7352 |
| 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 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,28,41,6,13,29,43,8,30,31,42)(3,14,16,46,18,36,26,57,17,47,51,64,19,52,53,60)(4,20,22,54,24,38,55,63,23,45,56,40,25,10,27,37)(9,32,33,49,34,58,61,48,35,59,39,21,44,50,62,15);(1,3,6,17)(2,9,13,35)(4,21,23,49)(5,26,29,53)(7,19,8,18)(10,36,38,52)(11,39,43,33)(12,44,30,34)(14,45,47,20)(15,24,48,25)(16,31,51,28)(22,32,56,59)(27,58,55,50)(37,60,63,57)(40,46,54,64)(41,61,42,62)Polycyclic presentation:
2^9|a^2=d,b^2=i,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=cg,d^b=degh,e^b=eh,f^b=fi,g^b=gi,d^c=dghi,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1758);