Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #345808 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #24176 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(10,15)(11,19)(12,13)(16,20)(17,18)(21,22);(2,5)(3,6)(7,8)(9,10,12,15,14,18,13,17)(11,20,22,23,21,16,19,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,19)(13,22)(14,21)(15,23)(17,24)(18,20);(1,2,4,5)(3,6,8,7)(10,17)(12,13)(15,18)(16,24)(19,22)(20,23)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253043);