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 | #46369 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4367 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)(5,6)(7,8)(9,10,15,19)(11,20,22,16)(12,21,23,17)(13,14,18,24);(1,2,5,6)(3,4,7,8)(9,11,15,22)(10,16,19,20)(12,13,23,18)(14,21,24,17)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,39)(38,40);(1,2,5,6)(3,8,7,4)(12,23)(13,18)(14,24)(17,21)(26,30)(27,34)(28,37)(39,40);(2,6)(3,7)(9,12,15,23)(10,17,19,21)(11,18,22,13)(14,16,24,20)(25,27)(26,31)(28,38)(29,34)(30,33)(32,40)(35,37)(36,39);(1,3,5,7)(2,4,6,8)(9,13,15,18)(10,14,19,24)(11,12,22,23)(16,17,20,21)(25,28)(26,32)(27,35)(29,37)(30,36)(31,39)(33,40)(34,38);(1,4)(2,7)(3,6)(5,8)(9,14)(10,18)(11,21)(12,16)(13,19)(15,24)(17,22)(20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965772);