Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | #60444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2578 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)(6,10,12,16)(11,13,18,19);(5,6,8,16,9,12,17,10)(7,13,15,11,14,19,20,18);(2,4);(5,7,8,15,9,14,17,20)(6,11,10,13,12,18,16,19);(7,14)(11,18)(13,19)(15,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,f^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72343918 |
| maximal normal subgroup of | #72343918 |
| maximal quotient of | #72343963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289113);