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 |
| prime radical | #59213 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4210 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)(11,19)(12,14)(13,17)(15,16)(18,22)(20,21)(23,24)(25,26)(27,32)(28,29)(30,40)(31,34)(33,38)(35,39)(36,37);(1,2,5,4)(3,6,7,8)(10,16)(12,22)(17,19)(20,23)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(2,4)(6,8)(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,24,18,21)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(3,7)(6,8)(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,21)(19,24)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,19,16,17)(12,20,22,23)(14,24,18,21)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36);(1,4,5,2)(3,8,7,6)(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,23,19,20)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8232623);