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×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 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)(6,8)(9,10)(11,12)(13,21)(14,15)(16,22)(17,20)(18,24)(19,23)(26,31)(28,38)(29,36)(32,35);(1,2,4,5)(3,6,7,8)(10,15)(11,17)(16,21)(18,23)(26,31)(32,35)(33,39)(34,40);(1,3,4,7)(2,6,5,8)(9,11,14,17)(10,12,15,20)(13,18,22,23)(16,19,21,24)(25,26)(27,35)(28,39)(29,34)(30,31)(32,37)(33,38)(36,40);(25,27,30,37)(26,32,31,35)(28,36,38,29)(33,40,39,34);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,40)(34,35);(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,19,20,24)(25,29,30,36)(26,34,31,40)(27,28,37,38)(32,33,35,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619517);