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 | |
4693 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)(3,5)(4,7)(6,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,33)(27,34,28,32)(29,35,39,38)(30,37,40,36);(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,31,28)(26,32,33,34)(29,36,39,37)(30,35,40,38);(1,3,6,7)(2,4,8,5)(12,20)(16,21)(18,24)(22,23)(26,33)(30,40)(32,34)(35,38);(1,4,6,5)(2,3,8,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,31,27)(26,34,33,32)(29,37,39,36)(30,38,40,35);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29)(26,35)(27,37)(28,36)(30,32)(31,39)(33,38)(34,40);(1,5,6,4)(2,7,8,3)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,40)(26,36,33,37)(27,35,28,38)(29,32,39,34)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805761);