Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,21,26,23,29,38,42,22,28,20,27,37,46,35,24)(19,30,32,44,34,45,39,43,33,41,31,48,40,47,36,25);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(18,24,28,42)(20,35)(21,38)(23,37)(25,41,43,30)(26,29,27,46)(31,36)(32,39)(34,40)(44,45,48,47);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)(17,19)(18,25)(20,36)(21,39)(22,33)(23,40)(24,41)(26,45)(27,47)(28,43)(29,48)(30,42)(31,35)(32,38)(34,37)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=ghi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=egh,f^c=fh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46204);