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 | |
4531 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,13)(14,16)(15,17)(19,20)(21,24)(22,23)(25,26,31,36)(27,30,35,32)(28,33,38,40)(29,34,37,39);(1,2,6,8)(3,5,7,4)(9,11,15,14)(10,16,17,18)(12,19,21,22)(13,23,24,20)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,34)(39,40);(1,3,6,7)(2,4,8,5)(10,17)(12,21)(16,18)(19,22)(25,28,31,38)(26,33,36,40)(27,37,35,29)(30,39,32,34);(1,4,6,5)(2,3,8,7)(9,12,15,21)(10,13,17,24)(11,19,14,22)(16,23,18,20)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,34)(36,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,12)(11,20)(14,23)(15,24)(16,22)(17,21)(18,19)(25,29,31,37)(26,34,36,39)(27,38,35,28)(30,40,32,33);(1,5,6,4)(2,7,8,3)(9,14,15,11)(10,18,17,16)(12,22,21,19)(13,20,24,23)(25,30)(26,35)(27,36)(28,39)(29,33)(31,32)(34,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091081);