Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1408 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1845 | C22 |
| prime radical | C23 | #353 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1845 ◁ G | |
| LECS | G ▷ #1845 ▷ 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)(11,20,23,34)(12,26,27,16)(13,18,21,17)(14,19)(15,22)(24,35,25,36)(28,32,37,33)(29,38,30,40)(31,39);(9,11,21,23,14,25,13,24)(10,15,19,31)(12,22,37,40,28,39,27,38)(16,20,33,36)(17,29,18,30)(26,34,32,35)Polycyclic presentation:
2^9|a^2=d,b^2=egi,c^2=gh,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=deg,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, 1781);