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 | #43443 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,7,5)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,24)(20,23)(25,26,31,35)(27,36,39,32)(28,38,37,33)(29,34,40,30);(1,3)(2,5)(4,8)(6,7)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,27,31,39)(26,32,35,36)(28,40,37,29)(30,38,34,33);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(13,22)(23,24)(28,37)(29,40)(30,34)(33,38);(2,4)(3,7)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,23)(21,22)(25,28)(26,33)(27,29)(30,36)(31,37)(32,34)(35,38)(39,40);(1,4,6,2)(3,8,7,5)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,29)(26,34)(27,37)(28,39)(30,35)(31,40)(32,38)(33,36);(1,5,6,8)(2,7,4,3)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,30,31,34)(26,29,35,40)(27,38,39,33)(28,36,37,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8965967);