Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94579 |
| 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:
(1,2,5,11,6,13,28,43,8,14,27,44,7,12,30,42)(3,15,17,45,18,46,53,60,19,48,52,63,29,47,25,57)(4,20,21,41,22,10,36,37,24,38,55,64,23,39,26,58)(9,31,32,49,33,51,61,50,35,59,40,56,34,54,62,16);(1,3)(2,9,12,33,14,35,13,34)(4,16)(5,25,27,53)(6,29)(7,18)(8,19)(10,15,20,46,39,48,38,47)(11,40,42,62,44,32,43,61)(17,28,52,30)(21,54,55,51)(22,56)(23,49)(24,50)(26,31,36,59)(37,63,41,57,58,45,64,60)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=g,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,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, 1777);