Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| 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 | #62471 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1635 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,24,29)(19,30,35,45)(20,21,26,40)(22,28,23,27)(25,33,46,34)(31,32,36,48)(37,42,41,39)(38,47,43,44);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,24,35)(18,25,29,46)(20,36,26,31)(21,38,40,43)(22,34,23,33)(27,30,28,45)(32,42,48,39)(37,47,41,44);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,24,26)(18,21,29,40)(19,31,35,36)(22,41,23,37)(25,43,46,38)(27,42,28,39)(30,32,45,48)(33,44,34,47);(17,21)(18,26)(19,32)(20,29)(22,39)(23,42)(24,40)(25,44)(27,41)(28,37)(30,36)(31,45)(33,38)(34,43)(35,48)(46,47);(3,10)(6,14)(11,16)(13,15)(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,37,26,41)(21,39,40,42)(25,45,46,30)(31,47,36,44)(32,38,48,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272529);