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 | |
4304 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)(11,18)(12,13)(14,15)(16,19)(17,24)(20,23)(21,22)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(1,2)(3,4)(5,7)(6,8)(10,15)(12,17)(16,18)(20,22)(26,31)(27,34)(33,36)(35,40);(1,3,5,8)(2,6,7,4)(10,15)(11,19)(12,17)(21,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3,5,8)(2,4,7,6)(9,11,14,19)(10,16,15,18)(12,22,17,20)(13,23,24,21)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,17)(10,13,15,24)(11,20,19,22)(16,21,18,23)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,21)(12,15)(14,24)(16,22)(18,20)(19,23)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9542148);