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 | |
4242 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,22)(23,24)(25,26,32,35)(27,30,34,40)(28,37,39,33)(29,31,38,36);(2,5)(6,7)(10,15)(11,19)(17,20)(18,24)(25,27,32,34)(26,30,35,40)(28,36,39,31)(29,33,38,37);(1,2)(3,6)(4,5)(7,8)(25,28,32,39)(26,33,35,37)(27,36,34,31)(29,30,38,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,16,15,21)(12,18,22,24)(13,20,23,17)(25,29)(26,31)(27,33)(28,30)(32,38)(34,37)(35,36)(39,40);(9,12)(10,17)(11,18)(13,16)(14,22)(15,20)(19,24)(21,23)(25,30)(26,34)(27,35)(28,38)(29,39)(31,33)(32,40)(36,37);(9,13)(10,18)(11,20)(12,21)(14,23)(15,24)(16,22)(17,19)(25,31)(26,29)(27,28)(30,33)(32,36)(34,39)(35,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388225);