Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23);(18,22)(21,23);(10,13)(14,16)(19,24)(21,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=fg,b^a=bf,c^a=cg,d^b=df200 conjugacy classes — view the full character table.
| maximal subgroup of | #72259468, #72259492, #72304042, … |
| maximal normal subgroup of | #72259468, #72259492, #72304042, … |
| maximal quotient of | #72259307 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420663);