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 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,32)(28,36)(29,39)(30,31)(33,38)(34,37)(35,40);(2,5)(3,7)(11,17)(15,21)(18,23)(22,24)(26,31)(32,40)(33,36)(34,39);(1,2)(3,6)(4,5)(7,8)(10,14)(15,21)(16,19)(22,24)(25,27,30,35)(26,32,31,40)(28,37,38,29)(33,39,36,34);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,30,38)(26,33,31,36)(27,29,35,37)(32,34,40,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29)(26,34)(27,28)(30,37)(31,39)(32,33)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39)Polycyclic presentation:
2^9|c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9275366);