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 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,31,33)(27,30,36,32)(28,38,39,34)(29,40,37,35);(1,3,5,8)(2,6,4,7)(25,27,31,36)(26,32,33,30)(28,37,39,29)(34,35,38,40);(10,14)(15,17)(16,21)(22,23)(26,33)(27,36)(28,39)(35,40);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,34)(27,37)(29,36)(30,40)(31,39)(32,35)(33,38);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,37)(26,35,33,40)(27,28,36,39)(30,34,32,38);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,32)(26,36,33,27)(28,40,39,35)(29,38,37,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7800610);