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 |
| prime radical | #52640 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,22)(20,24)(27,30)(32,34)(35,37)(39,40);(2,3)(6,7)(10,15)(12,20)(16,22)(18,21)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,3,5,2)(4,7,8,6)(10,15)(11,19)(17,24)(18,21)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,21)(13,17)(14,19)(15,22)(18,20)(23,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,12)(10,17)(11,18)(13,22)(14,20)(15,24)(16,23)(19,21)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,13,14,23)(10,18,15,21)(11,17,19,24)(12,16,20,22)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 9113840);