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 | |
4673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(2,5)(6,7)(10,14)(11,18)(16,20)(19,24)(26,30)(31,33)(32,36)(38,39);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(22,23)(27,34)(31,33)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019548);