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 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,16,19,23)(12,17,22,21)(13,20,24,18)(27,33)(28,35)(30,36)(31,38);(1,3,5,8)(2,6,7,4)(10,15)(12,22)(16,23)(18,20)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,2)(3,4)(5,7)(6,8)(10,15)(11,19)(17,21)(18,20)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,11,14,19)(10,16,15,23)(12,18,22,20)(13,21,24,17);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9117514);