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 | |
4699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10,13,14)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26,30,31)(27,35,36,32)(28,38,39,33)(29,37,40,34);(2,5)(3,8)(10,14)(12,20)(15,17)(19,24)(26,31)(29,40)(32,35)(33,38);(1,3)(2,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(26,31)(27,36)(33,38)(34,37);(1,4,7,6)(2,3,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,36)(26,32,31,35)(28,37,39,34)(29,33,40,38);(1,5,7,2)(3,4,8,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(1,6,7,4)(2,8,5,3)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29,30,40)(26,34,31,37)(27,33,36,38)(28,35,39,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662831);