Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6,8,7)(10,12)(13,14)(15,20)(16,19)(17,18);(5,6,8,7)(14,18)(19,20);(9,10)(11,12)(13,15,17,16)(14,19,18,20);(1,2,4,3)(5,6,8,7)(13,14)(15,19)(16,20)(17,18);(1,3,4,2)(5,7,8,6)(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=h,e^2=fh,b^a=bf,c^a=cg,e^a=ef,d^b=df176 conjugacy classes — view the full character table.
| maximal subgroup of | #72280784, #72280912 |
| maximal normal subgroup of | #72280784, #72280912 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420850);