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 | |
4276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,23)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,3,5,7)(2,4,8,6)(9,11)(10,16)(12,22)(13,19)(14,23)(15,24)(17,18)(20,21)(25,27,30,32)(26,31,34,28)(29,35,39,38)(33,40,37,36);(1,3)(2,6)(4,8)(5,7)(25,28,30,31)(26,32,34,27)(29,36,39,40)(33,38,37,35);(1,2,5,8)(3,4,7,6)(9,12,15,19)(10,17,20,14)(11,22,24,13)(16,18,21,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,22)(10,18,20,23)(11,19,24,12)(14,16,17,21)(25,27)(26,28)(29,38)(30,32)(31,34)(33,40)(35,39)(36,37);(1,4)(2,7)(3,8)(5,6)(9,14,15,17)(10,19,20,12)(11,23,24,18)(13,21,22,16)(25,29)(26,33)(27,35)(28,36)(30,39)(31,40)(32,38)(34,37)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gh,e^2=ghi,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, 9633603);