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 | #46318 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,16)(11,13,12,19)(14,22,23,17)(18,24,20,21)(25,26,31,30)(27,34,37,32)(28,29,39,33)(35,36,38,40);(1,3,4,7)(2,5,6,8)(9,11,15,12)(10,13,16,19)(14,18,23,20)(17,21,22,24)(25,27,31,37)(26,32,30,34)(28,38,39,35)(29,36,33,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(14,23)(21,24)(25,28,31,39)(26,33,30,29)(27,35,37,38)(32,36,34,40);(9,12,15,11)(10,13,16,19)(14,20,23,18)(17,21,22,24)(25,29)(26,28)(27,36)(30,39)(31,33)(32,38)(34,35)(37,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,12)(11,16)(14,21)(15,19)(17,20)(18,22)(23,24)(25,30,31,26)(27,32,37,34)(28,29,39,33)(35,36,38,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,22)(19,24)(25,29)(26,28)(27,36)(30,39)(31,33)(32,38)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837251);