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 | #47991 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,6,4)(25,26,31,34)(27,35,37,32)(28,38,39,29)(30,33,40,36);(1,2)(3,4)(5,7)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,31,37)(26,32,34,35)(28,33,39,36)(29,40,38,30);(1,3,5,6)(2,4,7,8)(10,14)(11,18)(12,21)(23,24)(28,39)(29,38)(30,40)(33,36);(1,4,5,8)(2,6,7,3)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,28,31,39)(26,29,34,38)(27,36,37,33)(30,35,40,32);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,29)(26,28)(27,30)(31,38)(32,36)(33,35)(34,39)(37,40);(1,3)(2,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,33)(27,29)(28,35)(31,40)(32,39)(34,36)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8965462);