Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18,21,20)(19,22,23,24)(25,26,31,36)(27,37,28,40)(29,35,30,33)(32,39,34,38);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,21,23)(18,22,20,24)(25,27,29,38,31,28,30,39)(26,32,33,37,36,34,35,40);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,20,21,18)(19,22,23,24)(26,33)(27,38)(28,39)(29,30)(32,34)(35,36);(3,7)(6,8)(9,12,13,11)(10,16,15,14)(25,28,30,38,31,27,29,39)(26,34,35,37,36,32,33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fghi,c^2=eg,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,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, 382065);