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 | |
4225 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)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,31,33)(27,34,29,28)(30,32,39,38)(35,40,37,36);(1,2)(3,4)(5,6)(7,8)(10,14)(15,17)(16,20)(23,24)(25,27,31,29)(26,28,33,34)(30,35,39,37)(32,36,38,40);(2,6)(3,7)(10,14)(11,18)(16,20)(19,21)(25,28,31,34)(26,27,33,29)(30,36,39,40)(32,35,38,37);(3,7)(4,8)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,26)(27,28)(29,34)(30,38)(31,33)(32,39)(35,40)(36,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,29,31,27)(26,28,33,34)(30,37,39,35)(32,36,38,40);(1,4)(2,3)(5,8)(6,7)(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,30,31,39)(26,32,33,38)(27,35,29,37)(28,36,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532661);