Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1845 |
| prime radical | C23 | #494 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,23,30,25,32,24,31)(19,33,35,44,37,27,36,48)(20,26,38,45,39,47,34,46)(21,40,41,28,42,29,22,43);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,22,34,23,35,21,20,25,37,41,38,24,36,42,39)(18,26,29,44,31,46,28,33,32,47,40,48,30,45,43,27);(2,7)(6,14)(8,12)(9,15)(17,20,21,36,24,34,22,37,25,39,42,35,23,38,41,19)(18,27,28,46,30,48,29,26,32,33,43,45,31,44,40,47)Polycyclic presentation:
2^9|a^2=f,b^2=dh,c^2=eg,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42970);