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 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(2,3)(6,7)(10,15)(12,23)(16,19)(18,21)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3,5,2)(4,7,8,6)(10,15)(11,20)(12,23)(13,24)(26,27)(31,33)(32,34)(38,39);(1,4)(2,6)(3,7)(5,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9397341);