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 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(12,20)(16,21)(18,24)(22,23)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(2,7)(3,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(28,36)(33,38)(35,37)(39,40);(1,3,6,4)(2,8,7,5)(10,14)(15,17)(16,21)(22,23)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,4,6,3)(2,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,5)(2,3)(4,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=h,e^2=gh,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, 8019655);