Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143949 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19616 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,20)(12,26)(13,28)(14,33)(15,19)(16,22)(17,30)(18,29)(21,34)(23,39)(24,35)(25,40)(27,38)(31,36)(32,37);(9,11,15,23)(10,16,19,35)(12,27,13,31)(14,34,29,25)(17,21,37,40)(18,38,33,36)(20,28,39,26)(22,32,24,30);(2,5)(3,8)(6,7)(9,10)(11,21)(12,28)(13,26)(14,33)(15,19)(16,31)(17,32)(18,29)(20,25)(22,38)(23,40)(24,36)(27,35)(30,37)(34,39)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fg,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51146);