Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,5)(4,12,13,8)(11,14,16,15)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,7)(3,10)(4,13)(14,15)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(36,44)(39,46)(42,48)(45,47);(2,7)(5,9)(8,12)(14,15)(18,23)(21,30)(24,27)(28,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,4,6,13)(2,8,7,12)(3,11,10,16)(5,14,9,15)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134669);