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 | #41032 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,14,17,22)(12,16,19,23)(18,21,20,24)(25,26,31,35)(27,29,36,32)(28,33,38,40)(30,34,39,37);(3,7)(5,8)(9,11,13,17)(10,14,15,22)(12,18,19,20)(16,21,23,24)(25,27)(26,32)(28,30)(29,35)(31,36)(33,37)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(12,19)(21,24)(25,28,31,38)(26,33,35,40)(27,30,36,39)(29,34,32,37);(9,12)(10,16)(11,18)(13,19)(14,21)(15,23)(17,20)(22,24)(25,29,31,32)(26,27,35,36)(28,37,38,34)(30,33,39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20)(13,17)(15,22)(16,24)(18,19)(21,23)(25,28)(26,33)(27,30)(29,34)(31,38)(32,37)(35,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,23)(20,24)(25,30)(26,34)(27,28)(29,33)(31,39)(32,40)(35,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8883513);