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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(2,3)(6,7)(10,14)(15,20)(16,21)(22,24)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(1,3,5,2)(4,7,8,6)(12,19)(16,21)(17,23)(22,24)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104533);