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 | #37764 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4364 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)(11,18)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,40)(35,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(25,27,31,35)(26,28,32,38)(29,39,40,30)(33,34,37,36);(2,3)(6,7)(10,16)(13,18)(17,20)(22,24)(26,32)(27,35)(29,40)(34,36);(4,8)(6,7)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,28)(26,27)(29,36)(30,33)(31,38)(32,35)(34,40)(37,39);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,29)(26,33)(27,30)(28,36)(31,40)(32,37)(34,38)(35,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,30)(26,34)(27,29)(28,37)(31,39)(32,36)(33,38)(35,40)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8519985);