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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(12,20)(15,22)(19,23)(21,24)(26,30)(31,37)(32,38)(39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23);(9,11,14,16)(10,13,17,18)(12,19,20,23)(15,21,22,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(9,12)(10,15)(11,19)(13,21)(14,20)(16,23)(17,22)(18,24);(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,22)(20,24)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7563800);