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 | |
4718 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,23)(21,22)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(2,5)(6,7)(9,11,14,19)(10,15,16,24)(12,22,18,20)(13,21,17,23)(27,28)(31,34)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(12,18)(21,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,4,8)(2,6,5,7)(9,12,14,18)(10,17,16,13)(11,20,19,22)(15,21,24,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,11)(10,15)(12,20)(13,23)(14,19)(16,24)(17,21)(18,22)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(9,13,14,17)(10,18,16,12)(11,21,19,23)(15,20,24,22)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954666);