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 | |
4442 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,23)(22,24)(25,26,31,35)(27,28,36,32)(29,37,39,33)(30,34,38,40);(1,2,5,6)(3,7,8,4)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,27)(26,32)(28,35)(29,34)(30,37)(31,36)(33,38)(39,40);(1,3)(2,4)(5,8)(6,7)(12,21)(15,20)(18,22)(23,24)(25,28,31,32)(26,27,35,36)(29,38,39,30)(33,40,37,34);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,18)(13,21)(14,23)(16,24)(17,20)(19,22)(25,29,31,39)(26,33,35,37)(27,34,36,40)(28,30,32,38);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,19)(14,17)(15,23)(18,21)(20,24)(25,30)(26,34)(27,33)(28,29)(31,38)(32,39)(35,40)(36,37);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,23)(20,21)(22,24)(25,27)(26,32)(28,35)(29,34)(30,37)(31,36)(33,38)(39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=hi,d^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9067121);