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 | |
4499 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,31,37)(27,38,39,32)(28,34,40,33)(29,35,36,30);(2,5)(6,7)(11,17)(14,21)(18,23)(22,24)(25,27)(26,32)(28,30)(29,33)(31,39)(34,36)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,31,40)(26,33,37,34)(27,30,39,35)(29,38,36,32);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,34)(27,35)(29,32)(30,39)(31,40)(33,37)(36,38);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29)(26,35)(27,34)(28,32)(30,37)(31,36)(33,39)(38,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,31,35)(26,36,37,29)(27,28,39,40)(32,34,38,33)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947665);