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)(9,10,14,17)(11,12,15,18)(13,16,22,24)(19,20,23,21)(25,26)(27,36)(28,29)(30,40)(31,33)(32,37)(34,39)(35,38);(1,3,6,7)(2,4,5,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,27,31,37)(26,32,33,36)(28,38,39,30)(29,40,34,35);(1,4,6,8)(2,7,5,3)(11,15)(12,18)(19,23)(20,21)(26,33)(27,37)(28,39)(35,40);(1,2,6,5)(3,8,7,4)(9,10)(11,18)(12,15)(13,16)(14,17)(19,21)(20,23)(22,24)(25,28)(26,34)(27,38)(29,33)(30,37)(31,39)(32,35)(36,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,18)(10,15,17,11)(13,21,22,20)(16,19,24,23)(25,29,31,34)(26,28,33,39)(27,35,37,40)(30,36,38,32);(1,5,6,2)(3,8,7,4)(9,13,14,22)(10,16,17,24)(11,19,15,23)(12,20,18,21)(25,30,31,38)(26,35,33,40)(27,28,37,39)(29,36,34,32)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=i,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, 9983106);