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 | |
4440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(3,7)(5,8)(10,14)(12,21)(15,17)(19,24)(27,33)(29,37)(31,34)(32,36);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(16,20)(19,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,30,26)(27,31,33,34)(29,36,37,32)(35,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8712042);