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 | #50006 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,17)(13,22)(14,15)(18,24)(19,23)(20,21)(25,26,31,29)(27,28,34,32)(30,37,38,33)(35,39,40,36);(1,3,5,8)(2,6,7,4)(9,10)(11,16)(12,20)(13,18)(14,15)(17,21)(19,23)(22,24)(26,29)(27,34)(33,37)(35,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,19)(12,21)(13,24)(25,27,31,34)(26,32,29,28)(30,35,38,40)(33,39,37,36);(1,4)(2,3)(5,6)(7,8)(9,11,14,19)(10,16,15,23)(12,22,21,18)(13,20,24,17)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(9,12)(10,17)(11,18)(13,23)(14,21)(15,20)(16,24)(19,22)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(9,13,14,24)(10,18,15,22)(11,20,19,17)(12,23,21,16)(25,30,31,38)(26,33,29,37)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8924361);