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 | |
4399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,15,17)(11,16,19,23)(12,14,21,24)(13,18,20,22)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,38,40,36);(1,3,6,5)(2,7,4,8)(9,11,15,19)(10,16,17,23)(12,20,21,13)(14,22,24,18)(26,31)(28,32)(34,37)(36,38);(10,17)(11,19)(12,21)(18,22)(25,27,30,33)(26,32,31,28)(29,35,39,40)(34,38,37,36);(2,4)(7,8)(9,12)(10,14)(11,20)(13,19)(15,21)(16,22)(17,24)(18,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,38)(36,37)(39,40);(1,4,6,2)(3,8,5,7)(9,13,15,20)(10,18,17,22)(11,21,19,12)(14,16,24,23)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,37,40);(1,5,6,3)(2,7,4,8)(9,14)(10,12)(11,22)(13,23)(15,24)(16,20)(17,21)(18,19)(25,29,30,39)(26,34,31,37)(27,35,33,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065849);