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 | |
4508 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)(3,7,5,8)(9,10)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(25,26,31,34)(27,30,33,39)(28,29,32,38)(35,36,37,40);(1,3,6,5)(2,7,4,8)(10,15)(11,20)(17,22)(18,21)(25,27)(26,30)(28,35)(29,36)(31,33)(32,37)(34,39)(38,40);(1,3,6,5)(2,8,4,7)(10,15)(13,24)(16,19)(17,22)(27,33)(30,39)(35,37)(36,40);(1,4,6,2)(3,7,5,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,38)(39,40);(1,5,6,3)(2,7,4,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,29)(26,32)(27,36)(28,34)(30,37)(31,38)(33,40)(35,39);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30,31,39)(26,33,34,27)(28,36,32,40)(29,37,38,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231324);