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 | #35195 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,21)(13,14)(15,16)(18,24)(19,20)(22,23)(25,26)(27,35)(28,29)(30,32)(31,34)(33,39)(36,38)(37,40);(2,6)(3,7)(10,16)(13,24)(17,21)(19,22)(25,27,31,30)(26,32,34,35)(28,36,39,37)(29,40,33,38);(1,3)(2,4)(5,7)(6,8)(9,11,15,12)(10,17,16,21)(13,19,24,22)(14,20,18,23)(25,28)(26,33)(27,36)(29,34)(30,37)(31,39)(32,38)(35,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,17,16,21)(13,22,24,19)(14,20,18,23)(25,28)(26,33)(27,37)(29,34)(30,36)(31,39)(32,40)(35,38);(9,13)(10,18)(11,19)(12,22)(14,16)(15,24)(17,23)(20,21)(25,29,31,33)(26,28,34,39)(27,38,30,40)(32,37,35,36);(9,14,15,18)(10,13,16,24)(11,20,12,23)(17,19,21,22)(25,30,31,27)(26,32,34,35)(28,37,39,36)(29,40,33,38)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412548);