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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,14)(13,18)(15,17)(19,24)(20,23)(21,22)(25,26,30,35)(27,36,38,31)(28,33,39,32)(29,40,37,34);(2,6)(3,8)(9,11,15,13)(10,16,17,18)(12,22,19,20)(14,21,24,23)(25,27,30,38)(26,31,35,36)(28,37,39,29)(32,34,33,40);(1,2)(3,7)(4,8)(5,6)(10,17)(11,13)(14,24)(20,22)(25,28,30,39)(26,32,35,33)(27,37,38,29)(31,34,36,40);(1,3,5,8)(2,7,6,4)(9,12,15,19)(10,14,17,24)(11,20,13,22)(16,23,18,21)(25,26)(27,36)(28,32)(29,34)(30,35)(31,38)(33,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,11)(10,18,17,16)(12,20,19,22)(14,23,24,21)(25,28)(26,33)(27,37)(29,38)(30,39)(31,40)(32,35)(34,36);(1,4)(2,3)(5,7)(6,8)(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,20)(18,22)(25,29,30,37)(26,34,35,40)(27,28,38,39)(31,33,36,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948212);