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 | |
4497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,4,5)(9,10,14,15)(11,16,18,13)(12,17,20,23)(19,21,24,22)(27,33)(31,34)(35,39)(38,40);(1,3)(2,7)(4,6)(5,8)(10,15)(11,18)(17,23)(19,24)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(2,8)(3,4)(12,20)(17,23)(19,24)(21,22)(26,28)(31,34)(32,36)(38,40);(1,3,6,4)(2,5,8,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,21)(20,24)(22,23)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,4,6,3)(2,5,8,7)(9,12)(10,17)(11,19)(13,22)(14,20)(15,23)(16,21)(18,24)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,5)(2,4)(3,8)(6,7)(9,13,14,16)(10,11,15,18)(12,21,20,22)(17,24,23,19)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=i,e^2=i,f^2=g,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, 9976139);