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 | |
4678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26,30,32)(27,28,34,36)(29,31,40,39)(33,35,37,38);(1,2,4,6)(3,5,7,8)(10,15)(12,16)(17,21)(20,22)(25,27)(26,28)(29,37)(30,34)(31,38)(32,36)(33,40)(35,39);(1,3)(2,5)(4,7)(6,8)(10,15)(11,18)(17,21)(19,24)(25,28,30,36)(26,27,32,34)(29,35,40,38)(31,33,39,37);(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,28)(26,27)(29,38)(30,36)(31,37)(32,34)(33,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,29)(26,31)(27,33)(28,35)(30,40)(32,39)(34,37)(36,38);(1,3)(2,5)(4,7)(6,8)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,26)(27,28)(29,39)(30,32)(31,40)(33,38)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9096692);