Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3482 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10)(11,15)(12,20,22,16)(13,17)(14,18)(19,24,21,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=fi,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,h^c=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293567);