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 | #39006 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 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)(11,15)(12,21)(13,23)(14,16)(17,22)(18,19)(20,24)(26,32)(30,40)(33,36)(34,38);(1,2,4,5)(3,6,7,8)(9,11,14,20)(10,15,16,24)(12,19,22,13)(17,23,21,18)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,40)(38,39);(10,16)(15,24)(17,21)(18,23)(25,27,31,28)(26,33,32,36)(29,35,39,37)(30,34,40,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,19)(13,20)(14,22)(15,23)(16,21)(18,24)(25,28,31,27)(26,33,32,36)(29,35,39,37)(30,38,40,34);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,29,31,39)(26,34,32,38)(27,37,28,35)(30,36,40,33);(9,13,14,19)(10,18,16,23)(11,12,20,22)(15,17,24,21)(25,30)(26,35)(27,34)(28,38)(29,36)(31,40)(32,37)(33,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550384);