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 | |
4771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,21)(14,17)(15,19)(16,23)(18,22)(20,24)(27,34)(29,31)(35,37)(38,40);(1,3,4,7)(2,5,6,8)(9,11,14,19)(10,15,17,12)(13,18,23,24)(16,22,21,20)(25,26)(27,29)(28,36)(30,33)(31,34)(32,39)(35,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,15,19,12)(13,16,23,21)(18,22,24,20)(25,27,30,34)(26,31,33,29)(28,35,39,37)(32,38,36,40);(9,10)(11,15)(12,19)(13,16)(14,17)(18,22)(20,24)(21,23)(25,27)(26,29)(28,37)(30,34)(31,33)(32,38)(35,39)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,15)(10,11,17,19)(13,22,23,20)(16,24,21,18)(25,28)(26,32)(27,35)(29,40)(30,39)(31,38)(33,36)(34,37);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,18)(12,20)(14,23)(15,22)(17,21)(19,24)(25,29)(26,27)(28,38)(30,31)(32,37)(33,34)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9096711);