Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(6,7)(9,10,12,11);(2,3)(14,16);(5,6,8,7)(9,10,12,11)(13,14,15,16);(5,7,8,6)(9,11,12,10);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72233675, D42×C22⋊C4, #72233680, … |
| maximal normal subgroup of | #72233675, D42×C22⋊C4, #72233680, … |
| maximal quotient of | #72252090, #72298106, #72298107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 441623);