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) = C23×C4 |
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 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,19,21,20,22,23,24);(1,2,4,8,5,7,3,6)(9,11)(10,13,14,15)(12,16);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=i,f^2=g,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, 3666);