Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
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)(3,14,16,44,17,36,51,62)(4,19,22,40)(6,11,29,41)(7,13,27,42)(8,30,31,15,33,48,60,20)(9,35,37,54)(12,28,43,21)(18,45,25,57,47,46,52,64)(23,39,26,58)(24,38,56,63)(32,55,59,49,34,50,61,53);(1,3)(2,8,11,32,13,33,12,34)(4,20)(5,25,27,52)(6,18)(7,17)(9,36,19,46,39,14,38,45)(10,31,41,59,42,60,43,61)(15,24)(16,29,51,28)(21,47)(22,55,56,50)(23,53)(26,48,35,30)(37,62,40,64,58,44,63,57)(49,54)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=gi,d^2=f,e^2=h,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, 1776);