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 | #46405 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,20)(11,16,13,18)(12,22,21,17)(14,19,23,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,20,18)(12,23,21,14)(17,19,22,24)(26,31)(27,32)(33,38)(34,39);(2,5)(6,8)(9,12,15,21)(10,17,20,22)(11,14,13,23)(16,24,18,19)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(3,7)(6,8)(9,13,15,11)(10,18,20,16)(12,14,21,23)(17,24,22,19)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,14)(10,19)(11,21)(12,13)(15,23)(16,17)(18,22)(20,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,18)(14,19)(15,20)(17,21)(23,24)(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=h,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9341918);