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 | |
4466 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)(11,19)(12,15)(13,17)(14,18)(16,22)(20,24)(21,23)(25,26)(27,30)(28,38)(29,34)(31,33)(32,35)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,35)(26,32,33,30)(28,36,34,40)(29,39,38,37);(1,2,4,6)(3,7,8,5)(13,21)(14,20)(17,23)(18,24)(26,33)(29,38)(30,32)(37,39);(3,8)(5,7)(9,12,15,10)(11,19,22,16)(13,17,21,23)(14,24,20,18)(25,28)(26,29)(27,36)(30,37)(31,34)(32,39)(33,38)(35,40);(1,3)(2,5)(4,8)(6,7)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29)(26,34)(27,37)(28,33)(30,40)(31,38)(32,36)(35,39);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30,31,32)(26,35,33,27)(28,37,34,39)(29,40,38,36)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=gh,e^2=gh,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, 8850527);