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 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,14)(12,15)(13,18)(16,22)(17,19)(20,24)(21,23)(25,26,30,31)(27,28,32,34)(29,37,39,33)(35,38,40,36);(1,3)(2,6)(4,7)(5,8)(10,11)(12,19)(16,18)(20,21)(25,26,30,31)(27,34,32,28)(29,33,39,37)(35,38,40,36);(10,11)(15,17)(16,18)(23,24)(26,31)(27,32)(33,37)(35,40);(2,7)(3,8)(9,11,14,10)(12,15,19,17)(13,18,22,16)(20,23,21,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,4)(2,8)(3,7)(5,6)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=hi,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, 10154878);