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 | |
4694 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)(11,16)(12,18)(13,20)(14,19)(15,17)(21,24)(22,23)(25,26,32,36)(27,37,39,33)(28,29,30,34)(31,35,38,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,16)(12,14)(13,17)(15,20)(18,19)(21,22)(23,24)(25,27)(26,33)(28,35)(29,31)(30,40)(32,39)(34,38)(36,37);(2,5)(6,7)(10,17)(11,20)(14,21)(18,23)(25,28,32,30)(26,34,36,29)(27,35,39,40)(31,33,38,37);(9,12)(10,18)(11,21)(13,19)(14,20)(15,22)(16,24)(17,23)(25,29)(26,28)(27,38)(30,36)(31,39)(32,34)(33,40)(35,37);(1,3,4,8)(2,6,5,7)(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,30,32,28)(26,29,36,34)(27,35,39,40)(31,33,38,37);(1,2)(3,6)(4,5)(7,8)(9,14,15,21)(10,19,17,24)(11,12,20,22)(13,23,16,18)(25,31,32,38)(26,35,36,40)(27,29,39,34)(28,37,30,33)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8122678);