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 | #35364 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4481 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)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,5)(6,7)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(27,28)(29,38)(31,34)(32,37);(1,2)(3,6)(4,5)(7,8)(11,18)(14,17)(19,24)(22,23)(25,27,30,28)(26,31,33,34)(29,36,38,35)(32,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40);(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=h,e^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, 8496144);