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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,19)(11,13,21,12)(14,17,24,22)(15,18,20,23)(25,26)(27,37)(28,30)(29,35)(31,32)(33,39)(34,36)(38,40);(1,2,5,3)(4,6,8,7)(9,11,16,21)(10,12,19,13)(14,20,24,15)(17,18,22,23)(26,32)(28,36)(29,38)(33,37);(2,3)(6,7)(9,12,16,13)(10,11,19,21)(14,23,24,18)(15,22,20,17)(25,27,31,39)(26,33,32,37)(28,38,36,29)(30,35,34,40);(2,3)(4,8)(9,13,16,12)(10,11,19,21)(14,18,24,23)(15,22,20,17)(25,28)(26,34)(27,38)(29,39)(30,32)(31,36)(33,40)(35,37);(1,3,5,2)(4,7,8,6)(9,14)(10,17)(11,20)(12,18)(13,23)(15,21)(16,24)(19,22)(25,29,31,38)(26,35,32,40)(27,36,39,28)(30,37,34,33);(1,4)(2,6)(3,7)(5,8)(9,15,16,20)(10,18,19,23)(11,14,21,24)(12,22,13,17)(25,30,31,34)(26,36,32,28)(27,35,39,40)(29,37,38,33)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012468);