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 | #59142 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4619 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)(3,7,8,5)(9,10)(11,17)(12,19)(13,16)(14,15)(18,24)(20,21)(22,23)(25,26)(27,30)(28,33)(29,38)(31,34)(32,37)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,12,15,17)(13,18,22,21)(16,20,23,24)(25,27,31,37)(26,32,34,30)(28,35,29,36)(33,40,38,39);(1,3)(2,5)(4,8)(6,7)(10,15)(12,17)(16,23)(20,24)(25,28,31,29)(26,33,34,38)(27,35,37,36)(30,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,29,31,28)(26,33,34,38)(27,36,37,35)(30,39,32,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,26)(27,30)(28,38)(29,33)(31,34)(32,37)(35,40)(36,39);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,30)(26,27)(28,39)(29,40)(31,32)(33,35)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,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, 8495126);