Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(5,6,8,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13,14,15)(12,16);(2,4)(5,6,8,7);(5,7,8,6)(9,10)(11,15)(12,14)(13,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,h^d=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D4×#99, #72311736, #72311781, … |
| maximal normal subgroup of | C4×D4×#99, #72311736, #72311781, … |
| maximal quotient of | #72312158, #72312525, #72313509 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264154);