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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(26,27)(31,33)(32,34)(39,40);(1,2,5,8)(3,4,7,6)(10,15)(12,16)(17,21)(20,22)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(17,21)(19,24);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|b^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9582120);