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 | |
4464 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)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26)(27,34)(28,38)(29,33)(30,31)(32,37)(35,36)(39,40);(1,2)(3,7)(4,5)(6,8)(11,17)(14,21)(18,23)(22,24)(26,30)(29,38)(32,34)(36,40);(2,5)(6,7)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,27)(26,32)(28,39)(29,40)(30,34)(31,37)(33,35)(36,38);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,29)(27,35)(30,38)(31,33)(32,36)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29,31,38)(26,33,30,28)(27,36,37,40)(32,39,34,35);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30,31,26)(27,34,37,32)(28,38,33,29)(35,40,39,36)Polycyclic presentation:
2^9|c^2=gh,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, 9144206);