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 | |
4618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,19)(11,15,18,13)(12,17,23,16)(20,22,21,24)(25,26)(27,32)(28,35)(29,31)(30,37)(33,39)(34,36)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11,14,18)(10,15,19,13)(12,21,23,20)(16,22,17,24)(25,27,31,35)(26,28,29,32)(30,34,39,38)(33,36,37,40);(1,3,5,8)(2,6,7,4)(9,12,14,23)(10,16,19,17)(11,20,18,21)(13,22,15,24)(25,28,31,32)(26,27,29,35)(30,36,39,40)(33,34,37,38);(3,8)(4,6)(9,12)(10,17)(11,21)(13,22)(14,23)(15,24)(16,19)(18,20)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,40,38,36);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,19)(14,18)(16,24)(17,22)(21,23)(25,27)(26,32)(28,29)(30,38)(31,35)(33,36)(34,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,15)(10,18,19,11)(12,22,23,24)(16,21,17,20)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9045640);