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 | |
4622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,13)(11,12,19,16)(14,17,22,21)(18,20,24,23);(1,2,6,5)(3,8,4,7)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,26,30,31)(27,32,36,38)(28,35,39,33)(29,34,40,37);(1,3,6,4)(2,7,5,8)(14,22)(17,21)(18,24)(20,23)(26,31)(29,40)(32,38)(33,35);(1,4,6,3)(2,8,5,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,38)(33,40)(34,39);(1,5,6,2)(3,8,4,7)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,28,30,39)(26,33,31,35)(27,34,36,37)(29,38,40,32);(9,14,15,22)(10,17,13,21)(11,18,19,24)(12,20,16,23)(25,29)(26,34)(27,35)(28,38)(30,40)(31,37)(32,39)(33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8273828);