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 | |
4648 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)(3,5)(4,7)(6,8)(10,15)(11,16)(17,21)(18,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,6,8)(3,5,7,4)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(28,36)(31,38)(34,40)(37,39);(1,3,6,7)(2,4,8,5)(11,16)(12,19)(18,23)(20,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,6,5)(2,3,8,7)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,5,6,4)(2,7,8,3)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669231);