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 | C2×#547 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,16,20)(11,13,17,12)(14,18,21,23)(15,19,24,22)(26,32)(28,33)(34,40)(35,38);(1,3,5,8)(2,6,7,4)(9,11,16,17)(10,12,20,13)(14,15,21,24)(18,22,23,19)(25,26,31,32)(27,33,37,28)(29,34,36,40)(30,35,39,38);(1,3)(2,4)(5,8)(6,7)(9,12,16,13)(10,17,20,11)(14,22,21,19)(15,18,24,23)(25,27,31,37)(26,33,32,28)(29,30,36,39)(34,35,40,38);(1,4)(2,3)(5,6)(7,8)(9,13,16,12)(10,11,20,17)(14,19,21,22)(15,23,24,18)(25,28)(26,27)(29,38)(30,34)(31,33)(32,37)(35,36)(39,40);(9,14)(10,18)(11,15)(12,22)(13,19)(16,21)(17,24)(20,23)(25,29)(26,34)(27,30)(28,38)(31,36)(32,40)(33,35)(37,39);(9,15)(10,19)(11,21)(12,18)(13,23)(14,17)(16,24)(20,22)(25,30)(26,35)(27,36)(28,34)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004918);