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 | |
4635 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26,31,30)(27,33,28,32)(29,34,37,38)(35,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27,31,28)(26,32,30,33)(29,35,37,36)(34,39,38,40);(1,2,5,7)(3,6,8,4)(10,14)(15,17)(16,20)(23,24);(2,7)(4,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28,31,27)(26,33,30,32)(29,35,37,36)(34,39,38,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,29,31,37)(26,34,30,38)(27,35,28,36)(32,39,33,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,30,31,26)(27,33,28,32)(29,34,37,38)(35,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050977);