Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94563 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7353 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1835 | C22 |
| prime radical | C23 | #353 |
| 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 | |
122 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)(3,14,16,46,17,48,51,64)(4,19,21,42)(6,11,29,43)(7,13,26,44)(8,31,32,49,34,59,41,15)(9,36,38,55)(12,27,45,30)(18,47,52,60,28,37,24,56)(20,33,50,61,53,35,58,62)(22,40,25,57)(23,39,54,63);(1,3)(2,8,13,34)(4,20)(5,24)(6,28)(7,17)(9,37,40,47)(10,41,44,32)(11,35,12,33)(14,19,48,39)(15,22)(16,27)(18,30)(21,31)(23,53)(25,58)(26,52)(29,51)(36,50)(38,60,57,56)(42,46,63,64)(43,61,45,62)(49,55)(54,59)Polycyclic presentation:
2^9|a^2=d,b^2=h,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,c^b=cg,d^b=deghi,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, 1759);