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 | |
4328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,18)(12,21)(13,16)(14,20)(15,17)(19,23)(22,24)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,3,6,5)(2,7,4,8)(9,11)(10,14)(12,22)(13,19)(15,20)(16,24)(17,18)(21,23)(25,26,31,28)(27,32,30,34)(29,33,38,36)(35,39,37,40);(9,12,15,13)(10,16,17,21)(11,19,20,22)(14,23,18,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2,6,4)(3,8,5,7)(9,11)(10,14)(12,19)(13,22)(15,20)(16,23)(17,18)(21,24)(25,28,31,26)(27,34,30,32)(29,33,38,36)(35,39,37,40);(1,4,6,2)(3,8,5,7)(9,13,15,12)(10,16,17,21)(11,19,20,22)(14,24,18,23)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,5,6,3)(2,7,4,8)(9,14)(10,11)(12,23)(13,24)(15,18)(16,19)(17,20)(21,22)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411691);