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 | |
4388 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)(9,10,15,14)(11,16,12,17)(13,18,22,21)(19,23,20,24)(25,26,31,28)(27,35,38,32)(29,39,37,33)(30,34,36,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,14,17)(13,19,22,20)(18,23,21,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,40)(34,39);(10,14)(16,17)(18,21)(23,24)(25,26,31,28)(27,35,38,32)(29,39,37,33)(30,34,36,40);(1,3,5,4)(2,6,8,7)(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,28,31,26)(27,35,38,32)(29,33,37,39)(30,34,36,40);(1,3,5,4)(2,6,8,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,31,37)(26,33,28,39)(27,36,38,30)(32,40,35,34);(1,4,5,3)(2,7,8,6)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,30,31,36)(26,34,28,40)(27,37,38,29)(32,39,35,33)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,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, 7982743);