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 | #42444 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4439 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)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26,31,35)(27,36,37,32)(28,29,33,39)(30,34,38,40);(1,2)(3,6)(4,7)(5,8)(25,27,31,37)(26,32,35,36)(28,38,33,30)(29,34,39,40);(10,14)(15,17)(16,22)(23,24)(27,37)(30,38)(32,36)(34,40);(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,28)(26,29)(27,38)(30,37)(31,33)(32,34)(35,39)(36,40);(1,3,5,4)(2,6,8,7)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,39)(36,38)(37,40);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,30,31,38)(26,34,35,40)(27,28,37,33)(29,32,39,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7558500);