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×#127 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,16,19,15)(13,20,21,17)(18,22,24,23)(25,26,31,28)(27,32,30,34)(29,33,38,36)(35,39,37,40);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,14)(13,17)(16,19)(18,22)(20,21)(23,24)(26,28)(32,34)(33,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,27,31,30)(26,32,28,34)(29,37,38,35)(33,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,12,14,10)(11,15,19,16)(13,17,21,20)(18,23,24,22)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,27)(26,34,28,32)(29,35,38,37)(33,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993568);