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 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26)(27,34)(28,38)(29,32)(30,39)(31,33)(35,40)(36,37);(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,31,29)(26,32,33,34)(28,36,39,35)(30,37,38,40);(1,3,6,5)(2,7,4,8)(9,11,14,12)(10,16,18,15)(13,19,21,20)(17,23,24,22)(25,28)(26,30)(27,35)(29,36)(31,39)(32,40)(33,38)(34,37);(1,3,6,5)(2,8,4,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,28)(26,30)(27,36)(29,35)(31,39)(32,37)(33,38)(34,40);(1,4,6,2)(3,8,5,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,29,31,27)(26,32,33,34)(28,36,39,35)(30,40,38,37);(1,5,6,3)(2,8,4,7)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,30)(26,28)(27,37)(29,40)(31,38)(32,36)(33,39)(34,35)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10318917);