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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(3,7)(6,8)(10,15)(12,19)(16,21)(20,22)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,21)(18,24)(27,34)(30,31)(35,40)(38,39);(9,11,14,17)(10,12,15,19)(13,18,23,24)(16,20,21,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,22)(14,19)(15,17)(16,24)(18,21)(20,23)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,23)(10,16,15,21)(11,18,17,24)(12,20,19,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40)Polycyclic presentation:
2^9|d^2=g,e^2=gh,f^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, 9074709);