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 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(5,7)(9,10,15,16)(11,17,21,24)(12,22,23,13)(14,20,19,18)(28,35)(32,37)(34,39)(38,40);(2,4)(3,8)(10,16)(13,22)(17,24)(18,20)(26,30)(27,33)(32,37)(34,39);(1,2,6,4)(3,7,8,5)(9,11,15,21)(10,17,16,24)(12,19,23,14)(13,20,22,18)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,6,8)(2,7,4,5)(9,12)(10,13)(11,19)(14,21)(15,23)(16,22)(17,20)(18,24);(1,4,6,2)(3,5,8,7)(9,13)(10,12)(11,20)(14,24)(15,22)(16,23)(17,19)(18,21)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,5)(2,3)(4,8)(6,7)(9,14,15,19)(10,18,16,20)(11,12,21,23)(13,24,22,17)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015945);