Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351670 |
| 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 | |
2632 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)(20,33)(21,28)(22,27)(23,26)(24,32)(25,35)(30,38)(31,41)(34,42)(36,39)(37,40)(43,46)(44,48)(45,47);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,39)(27,41)(28,29)(33,47)(34,44)(35,46)(37,45)(40,48)(42,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,23,34)(18,25,26,40)(19,30,31,44)(21,35,22,37)(24,38,39,48)(27,33,28,42)(29,43,41,47)(32,45,36,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,34)(25,40)(30,44)(33,42)(35,37)(38,48)(43,47)(45,46);(18,26)(19,31)(25,40)(27,28)(30,44)(32,36)(33,42)(45,46)Polycyclic presentation:
2^9|c^2=h,d^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,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, 6258885);