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 | |
4161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,16)(12,21)(13,18)(14,22)(15,17)(19,24)(20,23)(25,26,31,30)(27,35,38,32)(28,33,37,39)(29,34,36,40);(2,6)(3,7)(9,11,15,12)(10,16,17,21)(13,22,23,19)(14,20,24,18)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,34)(39,40);(1,3)(2,4)(5,7)(6,8)(10,17)(11,12)(18,20)(19,22)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,34,35,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,16,17,21)(13,22,23,19)(14,18,24,20);(9,13,15,23)(10,18,17,20)(11,19,12,22)(14,16,24,21)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,33,35,39);(9,14,15,24)(10,19,17,22)(11,20,12,18)(13,21,23,16)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,34,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=gi,f^2=gi,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, 9633393);