Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1196 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)(7,16,18,12)(8,17,11,19);(1,2)(3,4)(5,6)(7,17)(8,12)(9,13)(10,15)(11,16)(14,20)(18,19);(5,7,9,12,10,18,20,16)(6,11,13,17,15,8,14,19);(2,4)(5,8)(6,12)(7,14)(9,19)(10,11)(13,18)(15,16)(17,20)Polycyclic presentation:
2^9|a^2=h,c^2=efh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal quotient of | #72210005, #72239933, #72239934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296331);