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 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,31)(27,33,34,32)(28,29,37,38)(35,39,40,36);(1,2)(3,7)(4,5)(6,8)(10,11)(13,21)(15,17)(20,24)(26,31)(27,34)(29,38)(35,40);(2,5)(6,7)(12,19)(15,17)(20,24)(22,23)(26,31)(28,37)(32,33)(35,40);(2,5)(6,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,34)(26,32,31,33)(28,35,37,40)(29,36,38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,29)(27,35)(30,37)(31,38)(32,36)(33,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,28)(27,36)(30,38)(31,37)(32,35)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10229756);