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 | #43871 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4428 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)(3,8,4,6)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,23)(22,24)(25,26,31,35)(27,32,36,39)(28,33,38,37)(29,30,34,40);(3,4)(6,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,27)(26,32)(28,30)(29,37)(31,36)(33,34)(35,39)(38,40);(1,3,5,4)(2,6,7,8)(12,21)(15,20)(18,22)(23,24)(27,36)(28,38)(32,39)(33,37);(3,4)(6,8)(9,12,13,21)(10,15,17,20)(11,18,19,22)(14,23,16,24)(25,28)(26,33)(27,30)(29,39)(31,38)(32,34)(35,37)(36,40);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,22)(13,19)(14,17)(15,23)(18,21)(20,24)(25,29)(26,30)(27,37)(28,32)(31,34)(33,36)(35,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,23)(20,21)(22,24)(25,30)(26,34)(27,28)(29,35)(31,40)(32,33)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,e^2=h,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, 9034804);