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 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,2,5,4)(3,6,7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(26,31)(27,34)(33,36)(35,40);(2,4)(6,8)(9,12,14,15)(10,11,17,19)(13,21,22,20)(16,23,24,18)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(3,7)(6,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=gh,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, 8232665);