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 | C2×#546 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24)(25,26,31,33)(27,32,37,39)(28,34,35,29)(30,36,40,38);(3,7)(6,8)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(25,27)(26,32)(28,38)(29,40)(30,34)(31,37)(33,39)(35,36);(1,2,5,4)(3,6,7,8)(10,16)(13,19)(17,21)(18,24)(26,33)(27,37)(29,34)(36,38);(1,3,5,7)(2,6,4,8)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,28,31,35)(26,34,33,29)(27,36,37,38)(30,32,40,39);(1,4,5,2)(3,8,7,6)(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,29,31,34)(26,35,33,28)(27,30,37,40)(32,38,39,36);(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,30)(26,36)(27,34)(28,39)(29,37)(31,40)(32,35)(33,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510769);