Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
996 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)(10,13)(11,15)(14,16)(17,18,19,21,20,22,23,24);(1,3)(2,5,6,7)(4,8)(9,10)(11,16)(12,14)(13,15);(10,14)(13,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3602);