Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle, prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,18)(17,20);(6,9,11,10)(13,15)(14,17,18,19)(16,20)(21,22,23,26,24,27,29,32,25,28,30,33,31,34,35,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,c^b=cg152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 566);