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 | |
4625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,37)(29,39)(30,32)(33,34)(35,40)(36,38);(1,2,4,6)(3,5,7,8)(10,12)(11,17)(16,20)(18,23)(25,27,30,34)(26,31,32,33)(28,35,39,38)(29,36,37,40);(1,3)(2,5)(4,7)(6,8)(11,17)(13,21)(15,19)(16,20)(26,32)(27,34)(29,37)(35,38);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,33)(28,38)(29,36)(30,34)(31,32)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,29)(27,35)(30,39)(31,36)(32,37)(33,40)(34,38);(1,3)(2,5)(4,7)(6,8)(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29)(26,28)(27,36)(30,37)(31,35)(32,39)(33,38)(34,40)Polycyclic presentation:
2^9|b^2=h,e^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, 8840178);