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 | |
4027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(18,23)(19,26)(20,28)(30,31)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,2,6,7)(3,8,12,14)(4,11,13,9)(5,10,15,16)(17,18)(19,21)(20,29)(22,23)(24,28)(25,26)(27,30)(31,32)(35,42)(36,39)(37,46)(40,45);(2,7)(4,13)(8,14)(10,16)(17,19,22,26)(18,21,23,25)(20,27,28,32)(24,31,29,30)(33,35,38,42)(34,39,41,36)(37,47,46,43)(40,44,45,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,14)(11,15)(13,16)(17,20)(18,24)(19,27)(21,31)(22,28)(23,29)(25,30)(26,32)(33,36)(34,35)(37,44)(38,39)(40,43)(41,42)(45,47)(46,48);(1,4,6,13)(2,9,7,11)(3,10,12,16)(5,14,15,8)(17,19)(18,25)(20,27)(21,23)(22,26)(24,30)(28,32)(29,31)(33,37,38,46)(34,40,41,45)(35,43,42,47)(36,44,39,48);(1,5)(2,10)(3,11)(4,14)(6,15)(7,16)(8,13)(9,12)(17,21,22,25)(18,19,23,26)(20,30,28,31)(24,32,29,27)(33,34)(35,39)(36,42)(37,40)(38,41)(43,48)(44,47)(45,46)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,e^2=i,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, 8525283);