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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26,30,34)(27,35,36,31)(28,32,37,40)(29,33,39,38);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,31)(28,33)(29,32)(30,36)(34,35)(37,38)(39,40);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(2,7)(4,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,35)(28,32)(29,38)(30,34)(31,36)(33,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,37)(26,32,34,40)(27,33,36,38)(29,35,39,31);(1,4,5,6)(2,3,7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,39)(26,33,34,38)(27,32,36,40)(28,35,37,31)Polycyclic presentation:
2^9|a^2=hi,b^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8353293);