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 | |
4653 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)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(26,32)(27,28)(30,34)(37,39);(2,5)(6,7)(10,15)(11,17)(18,23)(19,24)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,33,32,36)(29,37,35,39)(30,38,34,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,40,34,38);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,40,36,38);(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,39,36,37)Polycyclic presentation:
2^9|c^2=h,d^2=ghi,e^2=ghi,f^2=h,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, 10426547);