Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2965 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,6,8,5,4,12,13)(3,9,10,16,11,7,14,15)(17,18,21,26,23,27,38,44,22,25,20,33,35,42,36,46)(19,28,31,41,32,43,37,40,30,47,29,45,39,48,34,24);(1,3,6,14,5,11,12,10)(2,7,13,15,4,9,8,16)(17,19)(18,24,25,40)(20,34)(21,37)(22,30)(23,39)(26,43,33,48)(27,45,42,41)(28,46,47,44)(29,36)(31,38)(32,35);(1,4)(2,5)(3,9)(6,8)(7,11)(10,15)(12,13)(14,16)(18,25)(20,21)(24,40)(27,42)(29,31)(34,37)(36,38)(41,45)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=egi,f^c=fg53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45608);