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 | |
4357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,17,13)(12,16,20,22)(19,24,23,21)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,2)(3,7)(4,6)(5,8)(10,15)(12,20)(13,18)(19,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,2,4,6)(3,5,8,7)(10,15)(11,17)(12,20)(21,24)(26,30)(27,34)(32,36)(35,40);(9,11,14,17)(10,13,15,18)(12,19,20,23)(16,21,22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,15)(12,21)(14,18)(16,23)(19,22)(20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9152487);