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 | #51888 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4348 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26,32,38)(27,35,29,33)(28,36,30,34)(31,37,40,39);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27,32,29)(26,33,38,35)(28,39,30,37)(31,36,40,34);(1,2,5,4)(3,6,8,7)(10,16)(15,18)(17,20)(23,24)(25,28,32,30)(26,34,38,36)(27,37,29,39)(31,33,40,35);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,29,32,27)(26,35,38,33)(28,37,30,39)(31,34,40,36);(9,12,14,22)(10,17,16,20)(11,19,13,21)(15,23,18,24)(25,30,32,28)(26,36,38,34)(27,37,29,39)(31,33,40,35);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,31)(26,37)(27,36)(28,33)(29,34)(30,35)(32,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=h,f^2=gh,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, 7862211);