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 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,26)(27,28)(29,37)(30,33)(31,34)(32,39)(35,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,19,17)(12,15,21,20)(18,22,24,23)(27,34)(28,31)(35,38)(36,40);(2,5)(3,7)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,27,30,34)(26,31,33,28)(29,38,39,35)(32,36,37,40);(3,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,40)(36,38)(37,39);(1,3,4,7)(2,6,5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28,30,31)(26,27,33,34)(29,36,39,40)(32,35,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,39)(31,40)(33,37)(34,38)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8864889);