Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366542 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,10)(5,14)(6,9)(8,11)(12,15)(13,16)(17,18,22,25)(19,27,29,43)(20,24,33,41)(21,34,37,26)(23,35,39,30)(28,45,46,40)(31,48,47,38)(32,44,36,42);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,22,29)(18,23,25,39)(20,31,33,47)(21,35,37,30)(24,40,41,45)(26,43,34,27)(28,36,46,32)(38,44,48,42);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,38)(29,39)(31,40)(32,42)(33,41)(34,37)(35,43)(36,44)(45,47)(46,48);(2,9)(7,15)(8,16)(10,14)(17,20,22,33)(18,24,25,41)(19,28,29,46)(21,36,37,32)(23,38,39,48)(26,44,34,42)(27,45,43,40)(30,47,35,31);(1,4,6,5)(2,10,9,14)(3,11,12,13)(7,8,15,16)(18,25)(19,29)(24,41)(26,34)(28,46)(30,35)(31,47)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273845);