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 | #48815 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,26,32,31)(27,28,34,30)(29,33,39,38)(35,36,40,37);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,30)(28,31)(29,35)(32,34)(33,37)(36,38)(39,40);(11,18)(12,20)(14,15)(16,23)(25,28,32,30)(26,27,31,34)(29,37,39,36)(33,40,38,35);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,32,39)(26,33,31,38)(27,35,34,40)(28,36,30,37);(1,2,5,4)(3,6,8,7)(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)(25,30,32,28)(26,34,31,27)(29,37,39,36)(33,40,38,35);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,31,32,26)(27,28,34,30)(29,38,39,33)(35,36,40,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7649789);