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 | #35683 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,15,19)(11,16,20,22)(12,14,18,24)(13,17,23,21)(25,26,31,34)(27,33,38,32)(28,30,29,39)(35,40,36,37);(1,2)(3,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,14)(15,20)(17,18)(19,22)(23,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(1,2)(3,5)(4,7)(6,8)(11,20)(13,23)(16,22)(17,21)(25,28,31,29)(26,30,34,39)(27,35,38,36)(32,37,33,40);(1,3,4,8)(2,5,7,6)(9,12)(10,14)(11,21)(13,16)(15,18)(17,20)(19,24)(22,23)(25,29,31,28)(26,30,34,39)(27,36,38,35)(32,37,33,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,14)(12,22)(15,23)(16,18)(19,21)(20,24)(25,27)(26,33)(28,36)(29,35)(30,37)(31,38)(32,34)(39,40);(1,2)(3,6)(4,7)(5,8)(9,14)(10,18)(11,13)(12,19)(15,24)(16,17)(20,23)(21,22)(25,30)(26,29)(27,37)(28,34)(31,39)(32,36)(33,35)(38,40)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973131);