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 | |
4511 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)(9,10,14,11)(12,15,19,17)(13,18,22,16)(20,24,21,23)(27,33)(31,34)(35,39)(38,40);(1,3)(2,7)(4,6)(5,8)(10,11)(12,19)(16,18)(20,21)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(2,8)(3,4)(10,11)(12,19)(16,18)(20,21)(26,28)(31,34)(32,36)(38,40);(1,3,6,4)(2,5,8,7)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(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)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(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)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=i,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, 9976122);