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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,19)(11,17,21,15)(12,22,23,16)(13,18,24,20);(3,7)(6,8)(9,11)(10,15)(12,20)(13,16)(14,21)(17,19)(18,23)(22,24)(25,26,30,31)(27,32,34,39)(28,29,33,38)(35,40,37,36);(1,2,5,4)(3,6,7,8)(12,23)(13,24)(16,22)(18,20)(26,31)(27,34)(29,38)(35,37);(9,12,14,23)(10,16,19,22)(11,20,21,18)(13,17,24,15)(25,27)(26,32)(28,37)(29,36)(30,34)(31,39)(33,35)(38,40);(1,3,5,7)(2,6,4,8)(9,11)(10,17)(12,18)(13,16)(14,21)(15,19)(20,23)(22,24)(25,28,30,33)(26,29,31,38)(27,35,34,37)(32,40,39,36);(1,4,5,2)(3,8,7,6)(9,13)(10,18)(11,16)(12,17)(14,24)(15,23)(19,20)(21,22)(25,29)(26,33)(27,36)(28,31)(30,38)(32,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826436);