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 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4423 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(2,5)(6,7)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(27,34)(31,38)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(10,16)(13,19)(17,21)(18,24)(25,27,30,34)(26,31,33,38)(28,35,32,39)(29,36,37,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510073);