Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131824 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61091 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
795 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 795 conjugacy classes of subgroups — too many to draw. Only its 116 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:
(1,2,5,7,4,3,8,6)(10,16)(11,21)(12,25)(13,29)(14,28)(17,34)(18,19)(20,30)(22,23)(24,27)(26,31)(32,39)(33,36)(37,38);(5,8)(6,7)(9,10,15,20)(11,17,24,38)(12,26,29,18)(13,30,25,16)(14,31,28,19)(21,35,27,40)(22,36,39,33)(23,34,32,37);(1,3)(2,4)(5,6)(7,8)(9,11)(10,17)(12,27)(13,22)(14,32)(15,24)(16,33)(18,35)(19,34)(20,38)(21,29)(23,28)(25,39)(26,40)(30,36)(31,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=g,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39072);