Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,7)(10,12)(13,14)(15,19)(16,17)(18,20);(1,2,4,3)(5,6,8,7)(14,17)(18,19);(9,10)(11,12);(13,14)(15,18)(16,17)(19,20);(1,3,4,2)(5,7,8,6)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df176 conjugacy classes — view the full character table.
| maximal subgroup of | #72233563, #72233567, #72233576, … |
| maximal normal subgroup of | #72233563, #72233567, #72233576, … |
| maximal quotient of | #72298404, #72298419, #72298491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420910);