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 | |
4503 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)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,24);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,13)(14,19,22,20)(18,23,24,21)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,2,5,4)(3,6,8,7)(10,17)(11,12)(14,22)(21,23)(26,31)(29,39)(32,40)(33,36);(9,12,15,11)(10,16,17,13)(14,20,22,19)(18,23,24,21)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,2,5,4)(3,6,8,7)(9,13,15,16)(10,12,17,11)(14,21,22,23)(18,20,24,19)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4,5,2)(3,7,8,6)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7675276);