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 | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
694 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)(11,21)(12,24)(13,27)(14,33)(15,18)(16,22)(17,25)(19,31)(20,30)(23,29)(26,28)(32,39)(34,37)(35,38)(36,40);(1,2,4,6)(3,8,7,5)(9,11,15,17)(10,16,18,12)(13,28,32,38)(14,34,30,23)(19,37,36,29)(20,26,33,35)(21,39,25,27)(22,40,24,31);(9,12,15,16)(10,17,18,11)(13,29,32,37)(14,35,30,26)(19,38,36,28)(20,23,33,34)(21,31,25,40)(22,27,24,39);(10,18)(11,17)(19,36)(20,33)(22,24)(23,34)(27,39)(28,38)Polycyclic presentation:
2^9|b^2=gh,c^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, 396244);