Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351787 |
| 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 | |
2714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,16)(17,18)(19,27)(20,24)(21,33)(22,25)(23,29)(26,36)(28,44)(30,41)(31,39)(32,42)(34,38)(35,40)(37,46)(43,45)(47,48);(1,2)(3,8)(4,9)(5,7)(6,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,38)(26,41)(27,33)(29,36)(31,45)(32,47)(35,46)(39,48)(40,43)(42,44);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,24)(19,28)(21,35)(22,31)(23,37)(25,39)(26,42)(27,43)(29,47)(30,45)(32,36)(33,40)(34,46)(38,48)(41,44);(3,12)(8,15)(11,13)(14,16)(18,25)(20,31)(23,38)(26,33)(27,41)(28,45)(32,35)(46,47);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,16,15,14)(18,25)(23,38)(24,39)(26,33)(27,41)(37,48)(40,42)(43,44)Polycyclic presentation:
2^9|e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259009);