Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14,16,21)(11,17)(12,19)(15,20,23,22)(18,24);(2,5)(6,8)(10,15,16,23)(11,18)(14,20,21,22)(17,24);(9,10,11,14,12,15,17,22,13,16,18,21,19,23,24,20);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225297, #72310153, #72310169, … |
| maximal normal subgroup of | #72225297, #72310153, #72310169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402515);