Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| 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 | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,17,19)(15,20,16,18);(5,7)(6,9,10,11)(8,12)(14,18)(15,16)(19,20);(1,3,4,2)(14,19)(18,20);(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dh,e^a=eh,g^a=ghi,d^b=dh,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72263997, #72264005, #72264012, … |
| maximal normal subgroup of | #72263997, #72264005, #72264012, … |
| maximal quotient of | C2×#182644, #72280054, #72280248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89840);