Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62991 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,24,21,23);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24);(2,5)(3,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(2,5)(3,7)(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503384);