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 | #45450 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(25,26)(27,29)(28,38)(30,33)(31,34)(32,36)(35,37)(39,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(18,21)(19,20)(25,27,31,32)(26,29,34,36)(28,39,33,35)(30,37,38,40);(2,3)(6,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,28,31,33)(26,30,34,38)(27,35,32,39)(29,40,36,37);(4,8)(6,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,26)(27,36)(28,30)(29,32)(31,34)(33,38)(35,37)(39,40);(1,3,5,2)(4,7,8,6)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,29,31,36)(26,32,34,27)(28,37,33,40)(30,35,38,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,30)(26,33)(27,37)(28,34)(29,35)(31,38)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=g,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, 8519407);