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 | |
4430 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)(3,5,7,8)(9,10)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24);(3,7)(5,8)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(25,26,30,31)(27,32,36,38)(28,35,37,33)(29,34,39,40);(1,2)(3,5)(4,6)(7,8)(10,16)(13,19)(17,21)(18,24)(26,31)(28,37)(32,38)(34,40);(1,3,4,7)(2,5,6,8)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,27)(26,32)(28,34)(29,33)(30,36)(31,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,28,30,37)(26,33,31,35)(27,34,36,40)(29,38,39,32);(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,29)(26,34)(27,35)(28,38)(30,39)(31,40)(32,37)(33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=h,e^2=i,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, 8525401);