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 | |
4477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,17,18,15)(12,16,21,22)(19,24,20,23)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(16,22)(19,20)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(2,3)(6,7)(10,14)(15,17)(16,22)(23,24)(29,38)(32,40)(34,37)(36,39);(4,8)(6,7)(9,11,13,18)(10,15,14,17)(12,20,21,19)(16,24,22,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,29,30,38)(26,32,33,40)(27,34,35,37)(28,36,31,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,21)(10,16,14,22)(11,19,18,20)(15,23,17,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=ghi,f^2=h,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, 8521390);