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 | |
4686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,18)(16,22)(17,19)(20,24)(21,23)(25,26,30,31)(27,33,34,28)(29,32,38,39)(35,37,40,36);(1,2)(3,7)(4,6)(5,8)(10,11)(12,19)(16,18)(20,21)(26,31)(27,34)(32,39)(35,40);(1,3)(2,5)(4,8)(6,7)(10,11)(15,17)(16,18)(23,24);(2,6)(5,7)(9,11,14,10)(12,15,19,17)(13,18,22,16)(20,23,21,24)(25,27,30,34)(26,28,31,33)(29,35,38,40)(32,36,39,37);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,28)(26,27)(29,37)(30,33)(31,34)(32,40)(35,39)(36,38);(1,3)(2,7)(4,8)(5,6)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gh,d^2=g,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, 10124975);