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 | #45457 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(2,6)(3,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(1,3)(2,4)(5,8)(6,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669319);