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 |
| prime radical | #34964 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,12)(11,16,19,21)(13,17,24,22)(14,18,20,23)(25,26,31,32)(27,29,34,39)(28,33,37,30)(35,38,40,36);(2,3)(6,7)(9,11)(10,16)(12,21)(13,14)(15,19)(17,18)(20,24)(22,23)(25,27)(26,29)(28,35)(30,36)(31,34)(32,39)(33,38)(37,40);(1,3,5,2)(4,7,8,6)(11,19)(14,20)(16,21)(18,23)(26,32)(27,34)(30,33)(35,40);(1,4)(2,6)(3,7)(5,8)(9,12,15,10)(11,16,19,21)(13,17,24,22)(14,23,20,18)(25,28)(26,33)(27,35)(29,38)(30,32)(31,37)(34,40)(36,39);(9,13)(10,17)(11,20)(12,22)(14,19)(15,24)(16,23)(18,21)(25,29,31,39)(26,27,32,34)(28,36,37,38)(30,35,33,40);(9,14,15,20)(10,18,12,23)(11,13,19,24)(16,17,21,22)(25,30,31,33)(26,28,32,37)(27,36,34,38)(29,35,39,40)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8309641);