Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| 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 | #51966 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1006 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)(9,10,12,15)(11,14,18,22)(13,21,20,16)(17,23,24,19);(2,5)(6,7)(9,10)(11,17)(12,15)(13,21)(14,19)(16,20)(18,24)(22,23);(9,11)(10,14)(12,18)(13,19,20,23)(15,22)(16,17,21,24);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,21)(17,24)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410207);