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 | |
4238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(10,15)(11,17)(16,23)(19,24)(25,26,32,36)(27,33,31,29)(28,37,39,34)(30,35,40,38);(2,5)(6,7)(9,10,14,15)(11,18,17,12)(13,16,22,23)(19,21,24,20)(25,27,32,31)(26,33,36,29)(28,38,39,35)(30,34,40,37);(1,2)(3,6)(4,5)(7,8)(25,28,32,39)(26,34,36,37)(27,35,31,38)(29,40,33,30);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,12,15,18)(13,19,22,24)(16,20,23,21)(25,29,32,33)(26,27,36,31)(28,30,39,40)(34,38,37,35);(9,12)(10,11)(13,21)(14,18)(15,17)(16,24)(19,23)(20,22)(25,30)(26,35)(27,37)(28,29)(31,34)(32,40)(33,39)(36,38);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,31,32,27)(26,33,36,29)(28,38,39,35)(30,37,40,34)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154738);