Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488933 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1629 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
943 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 943 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,29)(4,16)(5,21)(6,8)(7,13,22,32)(9,26)(10,19)(12,28,15,30)(17,25)(18,24,31,23)(20,27);(4,17)(5,19)(9,27)(10,21)(11,28)(12,15)(13,18)(16,25)(20,26)(23,24)(29,30)(31,32);(1,3)(2,7)(4,18)(5,12)(6,14)(8,22)(9,28)(10,23)(11,27)(13,17)(15,19)(16,29)(20,31)(21,24)(25,30)(26,32);(2,8)(7,22)(9,25)(10,21)(11,29)(16,27)(23,24)(28,30)Polycyclic presentation:
2^9|a^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215809, #72215883, #72219575, … |
| maximal normal subgroup of | #72215809, #72215883, #72219575, … |
| maximal quotient of | #72224621, #72226863 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396129);