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 | #46528 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,32,36)(27,34,29,31)(28,35,30,33)(37,39,38,40);(1,3,5,8)(2,6,4,7)(25,27,32,29)(26,31,36,34)(28,38,30,37)(33,40,35,39);(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,28,32,30)(26,33,36,35)(27,37,29,38)(31,39,34,40);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29,32,27)(26,34,36,31)(28,37,30,38)(33,39,35,40);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,30,32,28)(26,35,36,33)(27,37,29,38)(31,39,34,40);(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,31,32,34)(26,27,36,29)(28,39,30,40)(33,37,35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7808458);