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 | #52970 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4310 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)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(26,32)(28,33)(34,40)(36,38);(1,2)(3,8)(4,6)(5,7)(10,11)(15,17)(16,18)(22,23)(25,26,31,32)(27,33,29,28)(30,34,39,40)(35,38,37,36);(1,3,5,6)(2,4,7,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,38,40,36);(1,2)(3,4)(5,7)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,27)(29,32)(30,38)(31,33)(34,37)(35,40)(36,39);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,31,27)(26,28,32,33)(30,37,39,35)(34,36,40,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,39)(26,34,32,40)(27,35,29,37)(28,36,33,38)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8331838);