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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,16,19,23)(12,17,20,24)(13,21,22,18);(1,2,5,6)(3,7,4,8)(10,15)(12,20)(16,23)(18,21)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,5,4)(2,7,6,8)(10,15)(11,19)(17,24)(18,21)(26,31)(29,38)(32,39)(36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,17)(14,19)(15,23)(20,21)(22,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,12)(10,17)(11,18)(13,16)(14,20)(15,24)(19,21)(22,23)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,17)(12,16)(14,22)(15,21)(19,24)(20,23)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7938778);