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 | |
4530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(25,26,31,36)(27,37,40,32)(28,29,34,33)(30,35,39,38);(1,3,5,8)(2,6,7,4)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,31,40)(26,32,36,37)(28,35,34,38)(29,30,33,39);(1,2,5,7)(3,6,8,4)(10,14)(11,18)(12,21)(23,24)(28,34)(29,33)(30,39)(35,38);(2,7)(3,8)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,28)(26,33)(27,38)(29,36)(30,32)(31,34)(35,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,29,31,33)(26,34,36,28)(27,39,40,30)(32,35,37,38);(1,4,5,6)(2,8,7,3)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30,31,39)(26,35,36,38)(27,33,40,29)(28,32,34,37)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,e^2=gh,f^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, 8966142);