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 | |
4282 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)(9,10,15,19)(11,20,21,16)(12,22,23,17)(13,14,18,24)(26,31)(27,34)(29,37)(36,39);(2,6)(3,7)(9,11,15,21)(10,16,19,20)(12,18,23,13)(14,17,24,22)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,3)(2,4)(5,7)(6,8)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4)(2,3)(5,8)(6,7)(9,12,15,23)(10,17,19,22)(11,13,21,18)(14,20,24,16)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,13)(10,14)(11,12)(15,18)(16,17)(19,24)(20,22)(21,23)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(9,14)(10,18)(11,17)(12,20)(13,19)(15,24)(16,23)(21,22)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154812);