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 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,19)(11,20,22,16)(12,21,24,17)(13,14,18,23)(25,26,32,31)(27,30,36,35)(28,37,29,33)(34,40,38,39);(1,2,5,4)(3,7,8,6)(9,11,15,22)(10,16,19,20)(12,23,24,14)(13,21,18,17)(25,27)(26,30)(28,38)(29,34)(31,35)(32,36)(33,40)(37,39);(1,3,5,8)(2,6,4,7)(25,28,32,29)(26,33,31,37)(27,34,36,38)(30,39,35,40);(1,4,5,2)(3,7,8,6)(9,12,15,24)(10,17,19,21)(11,14,22,23)(13,16,18,20)(25,29,32,28)(26,33,31,37)(27,34,36,38)(30,40,35,39);(9,13)(10,14)(11,21)(12,16)(15,18)(17,22)(19,23)(20,24)(25,30)(26,27)(28,39)(29,40)(31,36)(32,35)(33,34)(37,38);(9,14)(10,18)(11,12)(13,19)(15,23)(16,17)(20,21)(22,24)(25,31,32,26)(27,35,36,30)(28,37,29,33)(34,40,38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8603947);