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 | |
4459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,30,34)(27,29,33,31)(28,32,38,36)(35,37,40,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(1,3)(2,4)(5,6)(7,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,26,30,34)(27,31,33,29)(28,32,38,36)(35,39,40,37);(3,6)(4,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,37)(39,40);(1,4,5,8)(2,6,7,3)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28,30,38)(26,32,34,36)(27,35,33,40)(29,37,31,39);(1,2)(3,8)(4,6)(5,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819778);