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 | |
3930 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)(9,15)(11,13)(33,34,38,42)(35,43,44,39)(36,40,46,47)(37,41,45,48);(1,2,6,7)(3,8,10,5)(4,9,12,15)(11,16,13,14)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(35,44)(37,45)(39,43)(41,48);(3,10)(4,12)(5,8)(9,15)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35,38,44)(34,39,42,43)(36,37,46,45)(40,48,47,41);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,12)(15,16)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36,38,46)(34,40,42,47)(35,45,44,37)(39,41,43,48);(1,4,6,12)(2,9,7,15)(3,11,10,13)(5,14,8,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,5,6,8)(2,3,7,10)(4,14,12,16)(9,11,15,13)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901534);