Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1782 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,14,16,10)(8,12)(9,13)(11,15,18,20)(17,19);(1,2,3,4)(6,10)(11,12)(13,14)(18,19);(5,7)(6,11,13,18)(8,15)(9,16)(10,12,14,19)(17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240802, #72257231, #72257242, … |
| maximal normal subgroup of | #72240802, #72257231, #72257242, … |
| maximal quotient of | #72240804 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53686);