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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,16)(11,12)(17,18,22,26)(19,27,29,23)(20,28,30,24)(21,25,32,31)(33,34,39,44)(35,45,46,40)(36,41,43,38)(37,42,48,47);(1,2,6,7)(3,8,10,5)(4,9,14,16)(11,13,12,15)(17,19,22,29)(18,23,26,27)(20,25,30,31)(21,24,32,28)(35,46)(37,48)(40,45)(42,47);(17,20,22,30)(18,24,26,28)(19,25,29,31)(21,23,32,27)(33,35,39,46)(34,40,44,45)(36,48,43,37)(38,47,41,42);(1,3)(2,8)(4,12)(5,7)(6,10)(9,15)(11,14)(13,16)(17,20)(18,24)(19,25)(21,27)(22,30)(23,32)(26,28)(29,31)(33,36,39,43)(34,41,44,38)(35,37,46,48)(40,47,45,42);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,21)(18,25)(19,28)(20,23)(22,32)(24,29)(26,31)(27,30)(33,37)(34,42)(35,36)(38,40)(39,48)(41,45)(43,46)(44,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,12,16,11)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,29)(28,30)(33,38,39,41)(34,43,44,36)(35,47,46,42)(37,45,48,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^2=ghi,f^2=gh,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, 10075666);