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 | |
4618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,34)(28,29)(30,38)(31,32)(33,39)(35,36)(37,40);(1,2,5,3)(4,6,8,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,29)(26,28,32,34)(30,35,39,37)(33,36,38,40);(2,3)(6,7)(10,14)(15,21)(16,19)(22,24)(26,32)(27,29)(33,38)(35,37);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,27)(29,32)(30,36)(31,34)(33,35)(37,38)(39,40);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,27)(26,28,32,34)(30,37,39,35)(33,36,38,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,39)(26,33,32,38)(27,35,29,37)(28,36,34,40)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8001921);