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 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,21)(22,23)(26,31)(27,34)(33,36)(35,40);(2,3)(6,7)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,39)(37,40);(1,2,5,3)(4,6,8,7)(10,12)(16,18)(17,20)(23,24);(9,12,15,10)(11,18,14,16)(13,20,21,17)(19,24,22,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,37,36,39);(1,3,5,2)(4,6,8,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,39,32,37);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,18,12,16)(13,19,21,22)(17,23,20,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755707);