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 | #36923 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,16,20)(13,21,23,17)(18,22,24,19)(25,26,32,30)(27,36,38,33)(28,34,29,39)(31,35,37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,22)(23,24)(25,27)(26,33)(28,31)(29,37)(30,36)(32,38)(34,40)(35,39);(1,3)(2,5)(4,8)(6,7)(10,15)(11,16)(17,21)(18,24)(25,28,32,29)(26,34,30,39)(27,31,38,37)(33,40,36,35);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,22)(23,24)(25,29,32,28)(26,34,30,39)(27,37,38,31)(33,40,36,35);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,22)(14,20)(17,18)(19,23)(21,24)(25,30,32,26)(27,36,38,33)(28,39,29,34)(31,35,37,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,23)(10,17,15,21)(11,18,16,24)(12,19,20,22)(25,31,32,37)(26,35,30,40)(27,28,38,29)(33,39,36,34)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=g,e^2=g,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^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9995739);