Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491437 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 40 points:
(2,5)(3,6)(7,8)(9,10)(11,19)(12,24)(13,16)(14,27)(15,22)(17,20)(18,25)(21,31)(23,38)(26,39)(28,36)(29,40)(30,35)(32,34)(33,37);(9,11,20,34,14,23,26,36,13,21,35,40,25,37,12,22)(10,15,24,33,18,29,30,31,16,28,39,38,27,32,17,19);(1,2,6,8,4,7,3,5)(10,16)(15,28)(17,30)(18,27)(19,31)(24,39)(29,32)(33,38);(11,21)(15,28)(19,31)(22,36)(23,37)(29,32)(33,38)(34,40)Polycyclic presentation:
2^9|b^2=eg,c^2=fhi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224818, #72224823, #72224824, … |
| maximal normal subgroup of | #72224818, #72224823, #72224824, … |
| maximal quotient of | #72225187, #72225191, #72225196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398667);