Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,14,10,15,7,12)(8,11,16,19,17,13,18,20);(6,11)(7,9)(8,17)(12,20)(13,15)(14,19);(2,4);(5,7,10,9)(6,12,15,14)(8,16,17,18)(11,19,13,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213392, #72213429, #72213443, … |
| maximal normal subgroup of | #72213392, #72213429, #72213443, … |
| maximal quotient of | C2×#493385, #72237543, #72237554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400581);