Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7535 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1409 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1844 | C22 |
| prime radical | C23 | #353 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1844 ◁ G | |
| LECS | G ▷ #1844 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,21,34)(12,24,25,32)(13,18)(14,17,27,30)(15,20)(16,26,31,40)(22,36,23,35)(28,39,33,38)(29,37);(9,11,14,23,13,22,27,21)(10,15,18,29)(12,20,25,38,26,37,40,39)(16,19,32,36)(17,28,30,33)(24,35,31,34)Polycyclic presentation:
2^9|a^2=d,b^2=egh,c^2=ghi,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=degi,d^c=dgh,e^d=ehi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1785);