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 | |
4398 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)(3,4)(5,8)(6,7)(9,10,16,18)(11,19,13,17)(12,22,14,15)(20,21,23,24)(28,36)(31,38)(34,40)(37,39);(1,3)(2,4)(5,6)(7,8)(9,11,16,13)(10,17,18,19)(12,20,14,23)(15,21,22,24)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(1,2,7,6)(3,5,8,4)(9,12,16,14)(10,15,18,22)(11,20,13,23)(17,21,19,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4,7,5)(2,8,6,3)(9,13,16,11)(10,17,18,19)(12,20,14,23)(15,24,22,21)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(1,5,7,4)(2,8,6,3)(9,14,16,12)(10,15,18,22)(11,20,13,23)(17,24,19,21)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,6,7,2)(3,5,8,4)(9,15)(10,14)(11,21)(12,18)(13,24)(16,22)(17,23)(19,20)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619991);