Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #412 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5,6,7)(9,10,13,16,14,11,12,15);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15);(9,10)(11,14)(12,15)(13,16);(9,12,14,13)(10,15,11,16)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72209982, #1042, #72226253, … |
| maximal normal subgroup of | #72209982, #1042, #72226253, … |
| maximal quotient of | C2×#186562, #72238828, #72238887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93758);