Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501156 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,17,18,24)(13,16,20,22)(15,23,21,19);(1,3,5,4)(2,6,8,7)(10,14)(11,18)(15,21)(16,22);(9,11,13,19)(10,15,16,17)(12,18,20,23)(14,21,22,24);(1,4,5,3)(2,7,8,6)(10,14)(15,21)(16,22)(17,24)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=f,d^2=ei,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410535);