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 | #37272 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,18)(12,15)(13,20)(14,16)(17,21)(19,23)(22,24)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(2,4)(7,8)(10,12)(13,21)(16,18)(19,22)(26,28)(27,33)(32,36)(35,39);(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,6,5)(2,7,4,8)(9,12,15,10)(11,18,14,16)(13,20,21,17)(19,24,22,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4,6,2)(3,8,5,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,5,6,3)(2,8,4,7)(9,14,15,11)(10,18,12,16)(13,22,21,19)(17,24,20,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015928);