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 | |
4329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(25,26,32,36)(27,37,40,33)(28,34,29,39)(30,35,38,31);(1,2,5,6)(3,7,8,4)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(25,27)(26,33)(28,38)(29,30)(31,39)(32,40)(34,35)(36,37);(1,3)(2,4)(5,8)(6,7)(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,28,32,29)(26,34,36,39)(27,38,40,30)(31,33,35,37);(2,6)(4,7)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,29,32,28)(26,34,36,39)(27,38,40,30)(31,37,35,33);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,30,32,38)(26,35,36,31)(27,29,40,28)(33,34,37,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,31,32,35)(26,30,36,38)(27,39,40,34)(28,33,29,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10133038);