Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138552 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,4,6,5,8,3,7)(10,16)(11,22)(12,25)(13,14)(17,26)(18,33)(19,30)(20,29)(21,31)(23,38)(24,27)(28,34)(35,40)(36,39);(2,6)(3,4)(7,8)(9,10,17,29,14,20,12,16,15,21,25,30,13,19,26,31)(11,18,32,40,22,34,23,28,24,35,37,33,27,36,38,39);(2,6)(3,4)(7,8)(9,11)(10,18)(12,23)(13,27)(14,22)(15,24)(16,28)(17,32)(19,36)(20,34)(21,35)(25,37)(26,38)(29,40)(30,33)(31,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45779);