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:
(2,6)(7,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,36)(28,38)(29,34)(30,32)(31,35)(33,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,30)(26,32,35,36)(28,29,37,39)(33,34,38,40);(1,3,5,4)(2,7,6,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,21)(22,24)(25,28)(26,33)(27,29)(30,39)(31,37)(32,34)(35,38)(36,40);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,29,31,39)(26,34,35,40)(27,37,30,28)(32,38,36,33);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30,31,27)(26,32,35,36)(28,39,37,29)(33,34,38,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,38)(39,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=g,e^2=h,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, 7564111);