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 | #41990 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,15,18)(11,17,12,16)(13,23,24,14)(19,22,21,20)(26,31)(28,37)(32,38)(35,40);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,18,17)(13,21,24,19)(14,20,23,22)(27,34)(32,38)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(25,26,30,31)(27,32,34,38)(28,33,37,29)(35,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,17,18,16)(13,21,24,19)(14,20,23,22)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,20)(18,23)(25,28)(26,33)(27,35)(29,31)(30,37)(32,39)(34,40)(36,38);(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,19)(18,24)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=gi,e^2=i,f^2=hi,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, 8416122);