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 | |
4469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24)(25,26,32,30)(27,33,29,28)(31,34,39,38)(35,40,37,36);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,27,32,29)(26,28,30,33)(31,35,39,37)(34,36,38,40);(1,2,5,4)(3,6,8,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,36)(32,33)(34,37)(35,38)(39,40);(1,2,5,4)(3,6,8,7)(9,12,14,21)(10,16,18,24)(11,20,22,13)(15,17,19,23)(25,29,32,27)(26,33,30,28)(31,37,39,35)(34,40,38,36);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,23)(14,18)(17,20)(19,22)(21,24)(25,30,32,26)(27,33,29,28)(31,34,39,38)(35,36,37,40);(9,13,14,20)(10,17,18,23)(11,21,22,12)(15,16,19,24)(25,31)(26,34)(27,35)(28,36)(29,37)(30,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562571);