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 | |
4493 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)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(2,5)(6,7)(10,14)(11,18)(16,20)(19,24)(28,36)(33,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(22,23)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(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=gh,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, 8019651);