Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2021 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)(8,12,16,10)(9,13)(14,18)(15,19,20,17);(1,2,3,4)(7,14)(8,16)(11,18)(17,19);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19);(5,7,9,14)(6,11,13,18)(8,15,16,20)(10,17,12,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256627, #72256628, #72256630, … |
| maximal normal subgroup of | #72256627, #72256628, #72256630, … |
| maximal quotient of | #72263714, #72263719, #72263723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401794);