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 | |
4387 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)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(1,2)(3,4)(5,6)(7,8)(10,14)(15,17)(16,20)(23,24)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(2,6)(3,7)(10,14)(15,17)(16,20)(23,24)(26,31)(27,33)(32,37)(34,40);(3,7)(4,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,28,30,36)(26,27,31,33)(29,38,39,35)(32,40,37,34);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38)Polycyclic presentation:
2^9|b^2=gh,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972788);