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 | |
4621 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)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26)(27,28)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,20)(19,21)(25,27)(26,29)(28,35)(30,31)(32,36)(33,37)(34,38)(39,40);(2,5)(6,8)(10,14)(15,17)(16,20)(23,24)(25,28,32,29)(26,27,35,36)(30,37,39,34)(31,33,40,38);(2,5)(3,7)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,29,32,28)(26,27,35,36)(30,34,39,37)(31,33,40,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,30)(26,33)(27,31)(28,34)(29,37)(32,39)(35,38)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,31)(26,34)(27,30)(28,33)(29,38)(32,40)(35,37)(36,39)Polycyclic presentation:
2^9|c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8985982);