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 | |
3916 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)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34)(35,40)(36,42)(37,43)(38,47)(39,41)(44,45)(46,48);(2,7)(4,9)(8,13)(12,14)(20,27)(24,30)(26,31)(28,32)(33,35,39,43)(34,40,41,37)(36,45,46,38)(42,44,48,47);(2,7)(3,10)(8,13)(11,16)(18,23)(19,25)(24,30)(26,31)(34,41)(35,43)(36,46)(44,47);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36)(34,42)(35,45)(37,47)(38,43)(39,46)(40,44)(41,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,37)(34,43)(35,41)(36,44)(38,48)(39,40)(42,45)(46,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38,39,45)(34,44,41,47)(35,36,43,46)(37,42,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123865);