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 | |
4643 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)(3,5)(4,8)(6,7)(10,14)(15,17)(16,21)(22,23);(2,7)(3,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,3,6,8)(2,4,7,5)(12,20)(16,21)(18,24)(22,23)(26,31)(29,39)(32,40)(33,36);(1,4,6,5)(2,3,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,2)(3,4)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,5,6,4)(2,8,7,3)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805189);