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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,16)(12,18)(13,19)(14,24)(15,17)(20,21)(22,23)(25,26)(27,35)(28,33)(29,36)(30,31)(32,37)(34,39)(38,40);(2,6)(3,7)(9,11,15,14)(10,16,17,24)(12,20,19,22)(13,23,18,21)(26,31)(28,38)(32,35)(34,36);(1,3)(2,4)(5,7)(6,8)(10,17)(12,19)(16,24)(20,22)(25,27,30,37)(26,32,31,35)(28,36,38,34)(29,33,39,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,19)(10,18,17,13)(11,20,14,22)(16,21,24,23)(25,27)(26,32)(28,34)(29,33)(30,37)(31,35)(36,38)(39,40);(9,13,15,18)(10,19,17,12)(11,21,14,23)(16,20,24,22)(25,28,30,38)(26,33,31,40)(27,36,37,34)(29,32,39,35);(9,14,15,11)(10,16,17,24)(12,22,19,20)(13,23,18,21)(25,29,30,39)(26,34,31,36)(27,33,37,40)(28,35,38,32)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=ghi,e^2=h,f^2=h,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, 10050693);