Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1646 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,21,22,23);(2,5)(6,7)(9,11,12,16)(10,13,14,15);(11,16)(13,15)(17,18)(19,23)(20,22)(21,24);(1,3,4,8)(2,6,5,7)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239596, #72239724, #72239725, … |
| maximal normal subgroup of | #72239596, #72239724, #72239725, … |
| maximal quotient of | #72239727, #72239732, #72257874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271264);