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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,18)(12,14)(13,16)(15,17)(19,23)(20,22)(21,24)(25,26,31,30)(27,32,35,37)(28,33,36,40)(29,38,39,34);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,17,18)(12,19,21,22)(14,20,24,23)(25,27,31,35)(26,32,30,37)(28,29,36,39)(33,38,40,34);(1,3,5,4)(2,6,8,7)(10,17)(14,24)(16,18)(20,23)(26,30)(27,35)(29,39)(33,40);(1,3,5,4)(2,6,8,7)(9,12)(10,14)(11,19)(13,22)(15,21)(16,20)(17,24)(18,23)(25,28,31,36)(26,33,30,40)(27,29,35,39)(32,38,37,34);(1,3,5,4)(2,6,8,7)(9,13,15,11)(10,16,17,18)(12,19,21,22)(14,23,24,20)(25,29)(26,34)(27,36)(28,35)(30,38)(31,39)(32,33)(37,40);(1,4,5,3)(2,7,8,6)(9,14)(10,12)(11,20)(13,23)(15,24)(16,19)(17,21)(18,22)(25,30,31,26)(27,37,35,32)(28,33,36,40)(29,38,39,34)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982761);