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 | |
4658 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)(13,21)(14,19)(15,20)(17,22)(23,24)(26,28)(27,33)(32,36)(35,37);(1,2)(3,8)(4,6)(5,7)(9,11,15,23)(10,16,20,24)(12,14,22,21)(13,18,19,17)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(1,3,5,6)(2,4,7,8)(9,12)(10,17)(11,21)(13,24)(14,23)(15,22)(16,19)(18,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,4,5,8)(2,3,7,6)(9,12)(10,18)(11,14)(13,24)(15,22)(16,19)(17,20)(21,23)(25,28,30,26)(27,34,33,31)(29,32,38,36)(35,39,37,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,14)(11,18)(12,24)(15,19)(16,22)(17,23)(20,21)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,22)(12,23)(13,20)(15,21)(16,17)(18,24)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8604729);