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 | |
4557 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)(4,6)(5,8)(9,10)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(10,15)(11,20)(17,22)(18,21)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(17,22)(18,21)(28,37)(32,36)(35,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,28,30,37)(26,32,33,36)(27,35,29,38)(31,39,34,40);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(9,13,14,24)(10,18,15,21)(11,17,20,22)(12,19,23,16)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)Polycyclic presentation:
2^9|b^2=gh,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8632017);