Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131664 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #62440 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
616 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 616 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:
(1,2,4,7,5,3,8,6)(9,10,13,16,15,20,12,26,14,19,27,32,28,36,29,18)(11,21,23,31,25,38,40,37,24,17,30,34,39,35,22,33);(1,3)(2,5)(4,6)(7,8)(10,16)(12,27)(13,29)(15,28)(17,31)(18,20)(19,32)(21,34)(22,23)(25,39)(26,36)(30,40)(33,35)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,22)(13,30)(14,24)(15,25)(16,31)(18,33)(19,21)(20,35)(23,27)(26,37)(28,39)(29,40)(32,34)(36,38)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,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, 38860);