Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351598 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,38)(31,39)(32,42)(35,40)(37,41)(43,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,23,32)(18,24,28,39)(20,30,37,45)(21,31,22,33)(25,38,40,48)(26,29,27,42)(34,47,41,43)(35,44,36,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,41)(27,34)(28,40)(29,43)(31,46)(32,45)(33,44)(39,48)(42,47);(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,29,39,42)(25,34,40,41)(30,44,45,46)(38,47,48,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,32)(24,39)(29,42)(30,45)(31,33)(38,48)(43,47)(44,46)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258824);