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 | |
4513 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)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26,30,33)(27,34,29,31)(28,32,36,37)(35,40,38,39);(1,2)(3,4)(5,6)(7,8)(10,14)(15,17)(16,20)(23,24)(25,27,30,29)(26,31,33,34)(28,35,36,38)(32,39,37,40);(2,6)(3,7)(10,14)(15,17)(16,20)(23,24)(28,36)(32,37)(35,38)(39,40);(3,7)(4,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,26)(27,31)(28,37)(29,34)(30,33)(32,36)(35,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28,30,36)(26,32,33,37)(27,35,29,38)(31,39,34,40);(1,4)(2,3)(5,8)(6,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,29,30,27)(26,31,33,34)(28,38,36,35)(32,39,37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972121);