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 | #48988 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4258 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(26,32)(27,28)(33,39)(35,36);(1,3,5,8)(2,6,4,7)(25,26,31,32)(27,34,28,29)(30,33,38,39)(35,40,36,37);(10,15)(13,22)(16,23)(20,24)(25,27,31,28)(26,29,32,34)(30,36,38,35)(33,40,39,37);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,31,27)(26,29,32,34)(30,35,38,36)(33,40,39,37);(1,2,5,4)(3,6,8,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,29)(26,27)(28,32)(30,37)(31,34)(33,35)(36,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,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, 7976703);