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 | |
4461 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)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,37)(38,40);(10,17)(12,18)(16,19)(20,22)(26,28)(27,33)(32,36)(35,40);(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,37,40,39);(1,2,5,4)(3,6,8,7)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,29)(26,32)(27,35)(28,36)(30,39)(31,38)(33,40)(34,37);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,30,31,34)(26,33,28,27)(29,37,38,39)(32,35,36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,e^2=gi,f^2=ghi,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, 7834332);