Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94579 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7366 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1835 | C22 |
| prime radical | C23 | #354 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1835 ◁ G | |
| LECS | G ▷ #1835 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
95 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)(3,14,16,43,17,45,50,62)(4,19,20,39)(6,11,27,40)(7,13,25,41)(8,29,30,47,32,58,38,15)(9,34,35,55)(12,26,42,28)(18,44,52,59,51,46,23,56)(21,37,24,57)(22,36,54,63)(31,49,60,64,33,53,61,48);(1,3)(2,8,11,31,13,32,12,33)(4,15)(5,23,25,52)(6,18)(7,17)(9,14,19,44,37,45,36,46)(10,38,40,61,41,30,42,60)(16,26,50,27)(20,53,54,49)(21,48)(22,47)(24,29,34,58)(28,51)(35,62,39,56,57,43,63,59)(55,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=g,d^2=f,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=deghi,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, 1775);