Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136733 |
| 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 | C2×#509 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,6,8)(9,10,12,19,14,20,26,16,15,21,13,18,28,38,29,33)(11,22,24,39,23,37,27,17,25,32,40,35,31,34,30,36);(1,3,5,8,4,2,6,7)(10,16,21,33)(11,23)(12,26)(13,29)(14,28)(17,34,36,37)(18,20,19,38)(22,35,32,39)(24,40)(25,31);(2,3)(7,8)(9,11)(10,17)(12,27)(13,30)(14,31)(15,25)(16,32)(18,37)(19,34)(20,39)(21,36)(22,33)(23,28)(24,26)(29,40)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319890, #72319904, #72319998, … |
| maximal normal subgroup of | #72319890, #72319904, #72319998, … |
| maximal quotient of | C2×#136734 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43930);