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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,22)(12,19)(13,18)(14,20)(15,21)(16,23)(17,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,23)(10,16,21,22)(12,20,24,13)(14,19,18,17)(26,27)(31,33)(32,34)(38,39);(9,12)(10,17)(11,20)(13,23)(14,16)(15,24)(18,22)(19,21)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,4)(2,7,6,8)(9,13,15,20)(10,18,21,14)(11,12,23,24)(16,17,22,19)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,3)(2,8,6,7)(9,12)(10,19)(11,20)(13,23)(14,22)(15,24)(16,18)(17,21);(9,14,15,18)(10,20,21,13)(11,19,23,17)(12,16,24,22)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=g,e^2=gh,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, 7564251);