Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C22 | C2×C43 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10,12,11)(18,21)(22,24);(2,3)(13,14,16,15);(5,6,7,8)(9,11,12,10)(13,15,16,14)(17,18,20,21)(19,22,23,24);(1,3,4,2)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=ef,b^a=be,c^a=cf,d^b=de200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60922);