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 | |
4430 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)(9,10,14,15)(11,16,19,22)(12,17,18,13)(20,23,24,21)(27,34)(32,35)(36,39)(37,40);(1,3,6,7)(2,4,5,8)(10,15)(11,19)(12,18)(21,23)(25,26,30,31)(27,35,34,32)(28,29,33,38)(36,40,39,37);(1,4,6,8)(2,7,5,3)(10,15)(11,19)(12,18)(21,23)(26,31)(28,33)(32,35)(36,39);(1,2,6,5)(3,8,7,4)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24)(25,27,30,34)(26,32,31,35)(28,36,33,39)(29,37,38,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,20)(13,15)(14,18)(16,23)(19,24)(21,22)(25,28)(26,29)(27,36)(30,33)(31,38)(32,37)(34,39)(35,40);(1,5,6,2)(3,8,7,4)(9,13,14,17)(10,18,15,12)(11,21,19,23)(16,24,22,20)(25,29)(26,33)(27,37)(28,31)(30,38)(32,39)(34,40)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=i,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, 9949749);