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 | |
4643 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,32,34)(27,28,30,29)(31,33,39,40)(35,36,38,37);(1,2,4,5)(3,6,7,8)(25,27,32,30)(26,29,34,28)(31,35,39,38)(33,37,40,36);(10,15)(13,22)(16,23)(20,24)(25,28,32,29)(26,27,34,30)(31,37,39,36)(33,38,40,35);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,29,32,28)(26,27,34,30)(31,36,39,37)(33,38,40,35);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,30,32,27)(26,28,34,29)(31,38,39,35)(33,36,40,37);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=g,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, 7982507);