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 | |
4343 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)(4,6)(5,8)(9,10)(11,12)(13,21)(14,17)(15,19)(16,22)(18,23)(20,24)(25,26,31,32)(27,33,30,28)(29,34,37,39)(35,40,38,36);(1,2,4,6)(3,5,8,7)(26,32)(28,33)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,27,31,30)(26,33,32,28)(29,35,37,38)(34,40,39,36);(1,3)(2,5)(4,8)(6,7)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,28)(26,27)(29,36)(30,32)(31,33)(34,35)(37,40)(38,39);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40);(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)(25,30,31,27)(26,28,32,33)(29,35,37,38)(34,40,39,36)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,f^2=ghi,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, 10441593);