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 | |
4576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,16)(11,17,19,12)(13,15,21,22)(18,23,24,20)(25,26,31,32)(27,35,38,30)(28,39,37,33)(29,40,36,34);(2,6)(3,7)(9,10)(11,17)(12,19)(13,15)(14,16)(18,23)(20,24)(21,22)(25,27,31,38)(26,30,32,35)(28,36,37,29)(33,34,39,40);(1,3)(2,4)(5,7)(6,8)(13,21)(15,22)(18,24)(20,23)(26,32)(30,35)(33,39)(34,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,19)(10,12,16,17)(13,18,21,24)(15,20,22,23)(25,28)(26,33)(27,36)(29,38)(30,34)(31,37)(32,39)(35,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,29,31,36)(26,34,32,40)(27,37,38,28)(30,33,35,39);(9,13,14,21)(10,15,16,22)(11,18,19,24)(12,20,17,23)(25,30,31,35)(26,27,32,38)(28,34,37,40)(29,33,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8309484);