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 | |
4441 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)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,26)(27,34)(28,36)(29,32)(30,31)(33,38)(35,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,40)(37,38);(1,2,4,6)(3,5,8,7)(10,14)(11,18)(16,20)(19,24)(25,26,31,30)(27,32,29,34)(28,36,38,33)(35,40,37,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,29,31,27)(26,32,30,34)(28,37,38,35)(33,39,36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,26)(27,32,29,34)(28,36,38,33)(35,39,37,40)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9152530);