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:
(1,2,5,11,7,12,26,39,6,13,27,41,8,29,30,40)(3,14,16,43,18,45,24,57,17,46,52,64,28,44,51,63)(4,19,20,53,22,37,54,62,21,42,55,38,23,10,25,36)(9,31,32,56,34,58,60,15,35,47,61,49,33,50,59,48);(1,3)(2,9,29,34,13,35,12,33)(4,15)(5,24,27,51)(6,17)(7,28)(8,18)(10,14,42,45,37,46,19,44)(11,32,40,60,41,61,39,59)(16,30,52,26)(20,50,55,58)(21,48)(22,56)(23,49)(25,31,54,47)(36,43,38,57,62,64,53,63)Polycyclic presentation:
2^9|a^2=d,b^2=ei,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, 1778);