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 |
| prime radical | #46804 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,31,28)(27,32,34,30)(29,36,37,33)(35,38,40,39);(1,2,5,6)(3,7,4,8)(25,27,31,34)(26,32,28,30)(29,35,37,40)(33,39,36,38);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,22)(18,23)(26,28)(27,34)(29,37)(38,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,26)(27,32,34,30)(29,36,37,33)(35,39,40,38);(1,4,5,3)(2,8,6,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,32)(26,34,28,27)(29,38,37,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621640);