Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C23×C8 |
| socle | C24 | C23×C4 |
| prime radical | C22×#120 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,16,19,17,15,18,20);(1,2,3,4)(13,15,16,20,17,14,18,19);(6,8);(10,12)Polycyclic presentation:
2^9|a^2=h,b^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72237122, #72237125, #72237127, … |
| maximal normal subgroup of | #72237122, #72237125, #72237127, … |
| maximal quotient of | #72239259, #72239285, #72239310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299812);