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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,6)(3,5,8,4)(9,10,15,12)(11,13,16,18)(14,20,23,17)(19,22,24,21)(25,26,30,34)(27,31,35,39)(28,32,38,37)(29,33,40,36);(1,3)(2,4)(5,6)(7,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(27,35)(28,38)(31,39)(32,37);(10,12)(11,16)(17,20)(19,24)(25,27,30,35)(26,31,34,39)(28,36,38,33)(29,37,40,32);(1,4,7,5)(2,8,6,3)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21);(1,5,7,4)(2,8,6,3)(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,28,30,38)(26,32,34,37)(27,36,35,33)(29,39,40,31);(1,6,7,2)(3,5,8,4)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,29)(26,33)(27,37)(28,31)(30,40)(32,35)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9758219);