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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,17)(12,19)(13,16)(14,15)(18,22)(20,21)(23,24)(25,26,31,29)(27,32,37,39)(28,34,38,33)(30,40,36,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,21)(14,19)(15,17)(16,22)(18,23)(20,24)(25,27,31,37)(26,32,29,39)(28,36,38,30)(33,40,34,35);(10,15)(12,17)(16,24)(20,22)(25,28,31,38)(26,33,29,34)(27,30,37,36)(32,35,39,40);(1,2,5,7)(3,6,8,4)(9,12)(10,11)(13,22)(14,17)(15,19)(16,21)(18,24)(20,23)(25,28)(26,34)(27,36)(29,33)(30,37)(31,38)(32,35)(39,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,16,15,24)(11,18,19,21)(12,20,17,22)(25,29,31,26)(27,32,37,39)(28,34,38,33)(30,35,36,40);(1,4,5,6)(2,3,7,8)(9,11)(10,12)(13,21)(14,19)(15,17)(16,22)(18,23)(20,24)(25,30,31,36)(26,35,29,40)(27,28,37,38)(32,33,39,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050620);