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 | |
4650 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)(3,4)(5,6)(7,8)(9,10,15,13)(11,17,19,12)(14,21,22,16)(18,20,24,23)(25,26,31,33)(27,37,38,32)(28,29,35,34)(30,40,39,36);(1,3,6,7)(2,4,5,8)(9,11,15,19)(10,12,13,17)(14,18,22,24)(16,20,21,23)(25,27)(26,32)(28,39)(29,40)(30,35)(31,38)(33,37)(34,36);(1,4,6,8)(2,7,5,3)(10,13)(12,17)(16,21)(20,23)(26,33)(27,38)(28,35)(36,40);(1,2,6,5)(3,8,7,4)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28,31,35)(26,34,33,29)(27,39,38,30)(32,36,37,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,10)(11,12,19,17)(14,16,22,21)(18,23,24,20)(25,29,31,34)(26,35,33,28)(27,36,38,40)(30,32,39,37);(1,5,6,2)(3,8,7,4)(9,14,15,22)(10,16,13,21)(11,18,19,24)(12,20,17,23)(25,30,31,39)(26,36,33,40)(27,35,38,28)(29,32,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9949763);