Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,10,14,11,7,15,13)(8,12,20,9,17,16,19,18);(5,7)(6,11)(8,18)(9,17)(10,13)(12,20)(14,15)(16,19);(2,4)(6,7)(9,18)(12,16)(13,14);(5,8)(6,12)(7,16)(9,14)(10,20)(11,17)(13,18)(15,19);(5,9,10,16,11,18,15,12)(6,8,14,20,7,17,13,19)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ci,d^b=df,f^b=fi,e^c=ei,f^d=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291057);