Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #299869 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19267 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,7,3)(4,5,8,6)(9,10,13,15)(11,17,20,24)(12,21,18,16)(14,22,23,19);(1,3,7,2)(4,5,8,6)(9,11,18,22,13,20,12,19)(10,14,21,24,15,23,16,17);(1,4,5,3,7,8,6,2)(10,15)(14,23)(16,21)(17,24);(9,11,18,22,13,20,12,19)(10,14,21,24,15,23,16,17)Polycyclic presentation:
2^9|a^2=hi,b^2=ehi,c^2=fh,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 207112);