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 |
| prime radical | #51780 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,14)(13,20)(15,19)(17,22)(18,23)(21,24)(25,26,31,32)(27,33,30,28)(29,34,37,39)(35,40,38,36);(9,10)(11,16)(12,14)(13,17)(15,19)(18,24)(20,22)(21,23)(26,32)(28,33)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,24)(21,22)(25,27,31,30)(26,33,32,28)(29,35,37,38)(34,40,39,36);(3,7)(6,8)(9,12,14,10)(11,16,19,15)(13,20,22,17)(18,24,21,23)(25,28)(26,27)(29,36)(30,32)(31,33)(34,35)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30,31,27)(26,28,32,33)(29,35,37,38)(34,40,39,36)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10387912);