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 | #43109 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,16,20,14)(12,18,21,24)(13,19,22,23)(25,26,31,33)(27,37,39,32)(28,29,35,34)(30,38,40,36);(3,7)(6,8)(9,11)(10,16)(12,13)(14,17)(15,20)(18,19)(21,22)(23,24)(25,27,31,39)(26,32,33,37)(28,30,35,40)(29,36,34,38);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(13,22)(18,24)(26,33)(29,34)(32,37)(36,38);(9,12)(10,18)(11,13)(14,23)(15,21)(16,19)(17,24)(20,22)(25,28)(26,34)(27,30)(29,33)(31,35)(32,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,19)(11,21)(12,20)(14,18)(15,22)(16,24)(17,23)(25,29,31,34)(26,35,33,28)(27,38,39,36)(30,37,40,32);(1,2)(3,6)(4,5)(7,8)(9,14)(10,20)(11,17)(12,19)(13,18)(15,16)(21,23)(22,24)(25,30)(26,36)(27,35)(28,39)(29,37)(31,40)(32,34)(33,38)Polycyclic presentation:
2^9|a^2=gi,b^2=g,e^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, 8972909);