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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,31,33)(27,34,30,32)(28,29,37,38)(35,39,40,36);(1,3,5,8)(2,6,4,7)(25,27,31,30)(26,32,33,34)(28,35,37,40)(29,36,38,39);(1,2,5,4)(3,6,8,7)(10,15)(13,16)(17,23)(21,24)(26,33)(27,30)(29,38)(35,40);(1,4,5,2)(3,7,8,6)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,28)(26,29)(27,35)(30,40)(31,37)(32,36)(33,38)(34,39);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20)(25,29)(26,28)(27,36)(30,39)(31,38)(32,35)(33,37)(34,40);(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,30,31,27)(26,32,33,34)(28,35,37,40)(29,39,38,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982701);