Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6)(7,8)(9,10,15,20,16,22,14,21)(11,23,25,17,27,36,26,37)(12,28,24,39,30,38,29,18)(13,31,32,40,34,35,33,19);(1,2)(3,4)(9,11,15,25,16,27,14,26)(10,17,20,36,22,37,21,23)(12,13,24,32,30,34,29,33)(18,31,28,40,39,35,38,19);(2,4)(9,12,16,30)(10,18,22,39)(11,13,27,34)(14,24,15,29)(17,35,37,31)(19,36,40,23)(20,38,21,28)(25,33,26,32);(6,8)(9,13,16,34)(10,19,22,40)(11,24,27,29)(12,25,30,26)(14,33,15,32)(17,18,37,39)(20,31,21,35)(23,38,36,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363634);