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 | |
4748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,14)(13,18)(15,17)(19,21)(20,23)(22,24)(25,26)(27,34)(28,37)(29,32)(30,38)(31,33)(35,36)(39,40);(2,4)(6,7)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,21,24,23)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(10,17)(11,12)(18,24)(19,20)(26,33)(28,38)(32,34)(35,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,24,21)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,21)(17,24)(25,29,31,27)(26,32,33,34)(28,39,38,35)(30,36,37,40);(1,3)(2,7)(4,6)(5,8)(9,14)(10,11)(12,17)(13,21)(15,16)(18,20)(19,24)(22,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9720160);