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 | |
4301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,40,36,38);(1,3,5,8)(2,7,6,4)(26,28)(27,34)(33,36)(35,37);(1,4)(2,8)(3,6)(5,7)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,28,31,26)(27,30,34,32)(29,36,39,33)(35,40,37,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,40,32,38);(1,2)(3,7)(4,8)(5,6)(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)(25,30)(26,27)(28,34)(29,38)(31,32)(33,37)(35,36)(39,40)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=gh,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9633707);