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 | |
4611 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)(11,16)(12,17)(13,19)(14,15)(18,23)(20,22)(21,24)(25,26)(27,34)(28,33)(29,30)(31,35)(32,39)(36,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,23)(21,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,39)(33,40)(34,35);(2,6)(3,7)(10,15)(12,22)(16,24)(18,19)(25,28,31,29)(26,33,35,30)(27,36,39,37)(32,40,34,38);(3,7)(4,8)(9,11,14,21)(10,16,15,24)(12,18,22,19)(13,17,23,20)(25,29,31,28)(26,33,35,30)(27,37,39,36)(32,40,34,38);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,27)(26,34)(28,37)(29,36)(30,38)(31,39)(32,35)(33,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,30,31,33)(26,29,35,28)(27,38,39,40)(32,37,34,36)Polycyclic presentation:
2^9|c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971330);