Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #273947 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18626 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,8,3,7,5,6,4)(9,10)(11,14)(12,19)(13,15)(16,22)(17,23)(18,20)(21,24);(1,3,7,4)(2,6,5,8)(10,15)(11,17)(13,23)(16,24)(18,19)(20,22);(1,4,7,3)(2,8,5,6)(9,11,14,10)(12,18,21,16)(13,15,23,17)(19,22,24,20);(1,5)(2,7)(3,8)(4,6)(9,12)(10,16)(11,18)(13,22)(14,21)(15,24)(17,19)(20,23)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=i,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 181263);