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 |
| prime radical | #42477 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,20)(11,21,22,16)(12,18,23,17)(13,14,19,24)(28,36)(31,38)(34,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,20,21)(12,19,23,13)(14,17,24,18)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,24)(18,20)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,2,5,4)(3,6,8,7)(9,13,15,19)(10,14,20,24)(11,12,22,23)(16,17,21,18)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,19)(13,22)(14,16)(15,23)(17,20)(21,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,17)(12,21)(13,20)(15,24)(16,23)(18,22)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562893);