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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10,13,14)(11,17,19,15)(12,16,22,20)(18,24,21,23);(1,2)(3,4)(5,7)(6,8)(10,14)(15,17)(16,20)(23,24)(25,26,30,31)(27,29,33,38)(28,32,36,39)(34,37,40,35);(1,3,5,8)(2,6,7,4)(10,14)(15,17)(16,20)(23,24)(26,31)(27,33)(32,39)(34,40);(2,7)(3,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,39)(36,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,2)(3,6)(4,8)(5,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,29)(26,33)(27,31)(28,35)(30,38)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616457);