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 | |
4448 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)(3,5)(4,8)(6,7)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(27,33)(28,36)(31,34)(32,37);(2,7)(3,8)(10,14)(11,17)(16,21)(19,23)(25,26)(27,34)(28,37)(29,30)(31,33)(32,36)(35,39)(38,40);(1,3,6,8)(2,4,7,5)(11,17)(12,20)(15,18)(16,21)(26,30)(27,33)(32,37)(35,38);(1,4,6,5)(2,3,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27)(26,31)(28,38)(29,33)(30,34)(32,40)(35,36)(37,39);(1,2)(3,4)(5,8)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,39)(33,38)(34,40);(1,5,6,4)(2,8,7,3)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840274);