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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4036 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,31)(28,32)(29,30)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(2,4)(3,10)(8,12)(11,13)(17,19,23,28)(18,24,25,32)(20,29,26,27)(21,30,31,22)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,4)(7,9)(8,12)(15,16)(18,25)(19,28)(21,31)(27,29)(35,40)(37,45)(43,46)(44,48);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,20,23,26)(18,21,25,31)(19,27,28,29)(22,32,30,24)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21)(18,26)(19,22)(20,25)(23,31)(24,29)(27,32)(28,30)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22)(18,27)(19,21)(20,24)(23,30)(25,29)(26,32)(28,31)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528819);