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 | #42533 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,16)(11,19,13,17)(12,18,22,14)(20,24,23,21)(25,26,31,30)(27,34,28,32)(29,33,37,38)(35,40,36,39);(1,2,6,5)(3,8,4,7)(10,16)(12,22)(17,19)(20,23)(25,27,31,28)(26,32,30,34)(29,35,37,36)(33,39,38,40);(1,3,6,4)(2,7,5,8)(9,11,15,13)(10,17,16,19)(12,23,22,20)(14,21,18,24)(25,26,31,30)(27,32,28,34)(29,33,37,38)(35,39,36,40);(1,4,6,3)(2,8,5,7)(9,12,15,22)(10,18,16,14)(11,20,13,23)(17,21,19,24)(25,28,31,27)(26,32,30,34)(29,36,37,35)(33,39,38,40);(1,5,6,2)(3,8,4,7)(9,13,15,11)(10,19,16,17)(12,23,22,20)(14,21,18,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,23,19,20)(25,30,31,26)(27,34,28,32)(29,33,37,38)(35,39,36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8655089);