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 | #42508 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,16,19)(11,18,14,17)(12,15,13,23)(20,24,21,22)(27,35)(28,37)(31,38)(32,40);(1,2,6,5)(3,8,4,7)(9,11,16,14)(10,17,19,18)(12,20,13,21)(15,22,23,24)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,36,39);(1,3,6,4)(2,7,5,8)(9,12,16,13)(10,15,19,23)(11,20,14,21)(17,22,18,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(1,4,6,3)(2,8,5,7)(9,13,16,12)(10,15,19,23)(11,21,14,20)(17,22,18,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,5,6,2)(3,8,4,7)(9,14,16,11)(10,18,19,17)(12,21,13,20)(15,24,23,22)(25,28,30,37)(26,32,34,40)(27,36,35,29)(31,39,38,33);(9,15)(10,13)(11,22)(12,19)(14,24)(16,23)(17,21)(18,20)(25,29,30,36)(26,33,34,39)(27,37,35,28)(31,40,38,32)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8370864);