Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,15,16,10)(9,12)(13,19,20,17)(14,18);(1,2,3,4)(8,16)(13,20);(6,10)(11,17)(12,15)(18,19);(5,7,9,14)(6,11,12,18)(8,13,16,20)(10,17,15,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fhi,e^c=eh,g^c=gh,f^e=fi,g^f=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231756, #72234768, #72234782, … |
| maximal normal subgroup of | #72231756, #72234768, #72234782, … |
| maximal quotient of | #72231753, #72234771, #72251723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401767);