Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,21)(11,22,25,38)(12,20,14,18)(13,16,29,32)(17,33,35,24)(19,26,31,27)(23,39,30,36)(28,37,40,34);(1,2,4,6)(3,8,7,5)(10,16)(12,14)(13,27)(17,22)(18,31)(19,20)(21,32)(23,30)(24,28)(26,29)(33,40)(34,39)(35,38)(36,37);(9,11)(10,17)(12,23)(13,28)(14,30)(15,25)(16,22)(18,34)(19,36)(20,37)(21,35)(24,27)(26,33)(29,40)(31,39)(32,38);(9,12,15,14)(10,18,21,20)(11,23,25,30)(13,26,29,27)(16,19,32,31)(17,34,35,37)(22,36,38,39)(24,28,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396248);