Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,8)(9,10)(11,12)(13,14,18,19,17,15,16,20);(1,2,3,4)(13,15,18,20,17,14,16,19);(5,6,7,8)(13,16,17,18)(14,19,15,20);(10,12)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72237122, #72237132, #72237137, … |
| maximal normal subgroup of | #72237122, #72237132, #72237137, … |
| maximal quotient of | #72239254, #72239260, #72239306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300150);