Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127645 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,19,12,16)(11,13,21,23,17,24,20,22);(2,6)(3,4)(7,8)(9,11,13,16,14,20,23,19,15,17,24,18,12,21,22,10);(1,3,5,4)(2,7,8,6)(9,12,15,14)(10,16,19,18)(11,20,17,21)(13,22,24,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=g,e^2=i,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ch,d^b=dh,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72342557, #72342560, #72342563, … |
| maximal normal subgroup of | #72342557, #72342560, #72342563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34985);