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 | #41416 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 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)(3,7,5,8)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3,6,5)(2,7,4,8)(9,10,14,19)(11,17,20,15)(12,16,22,23)(13,18,24,21)(26,31)(27,32)(33,38)(34,39);(2,4)(3,5)(9,11,14,20)(10,15,19,17)(12,18,22,21)(13,23,24,16)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(1,3,6,5)(2,7,4,8)(9,12)(10,16)(11,18)(13,15)(14,22)(17,24)(19,23)(20,21)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,4,6,2)(3,7,5,8)(9,11)(10,17)(12,21)(13,16)(14,20)(15,19)(18,22)(23,24);(1,5,6,3)(2,7,4,8)(9,13)(10,18)(11,16)(12,15)(14,24)(17,22)(19,21)(20,23)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=hi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9405496);