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 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,15,17)(11,16,13,21)(12,18,22,23)(14,19,24,20)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,2,5,8)(3,6,4,7)(9,11,15,13)(10,16,17,21)(12,19,22,20)(14,18,24,23)(26,31)(27,33)(32,38)(34,39);(1,3,5,4)(2,6,8,7)(10,17)(11,13)(18,23)(19,20)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,12)(10,18)(11,20)(13,19)(14,16)(15,22)(17,23)(21,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,4,5,3)(2,7,8,6)(9,13,15,11)(10,16,17,21)(12,19,22,20)(14,23,24,18)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40);(9,14)(10,19)(11,18)(12,21)(13,23)(15,24)(16,22)(17,20)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8436105);