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 | |
4862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(11,17)(15,18)(19,23)(22,24)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2,4,6)(3,5,7,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(26,30)(31,33)(32,36)(38,39);(1,3)(2,5)(4,7)(6,8)(10,14)(15,18)(16,20)(22,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658971);