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 | |
4415 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)(3,7)(4,5)(6,8)(11,17)(15,18)(19,23)(22,24)(25,26,30,35)(27,33,37,31)(28,32,38,40)(29,34,39,36);(2,5)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,37)(26,31,35,33)(28,36,38,34)(29,32,39,40);(2,5)(6,7)(10,14)(15,18)(16,20)(22,24)(25,28,30,38)(26,32,35,40)(27,36,37,34)(29,31,39,33);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27)(26,33)(28,36)(29,32)(30,37)(31,35)(34,38)(39,40);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,28)(26,32)(27,36)(29,33)(30,38)(31,39)(34,37)(35,40);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,34)(27,32)(28,31)(30,39)(33,38)(35,36)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9945672);