Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 32 points:
(1,2,4,10,6,11,17,26)(3,8,13,24,15,9,18,25)(5,7,16,22,19,23,29,32)(12,21,27,31,14,20,28,30);(2,7)(4,16)(8,20)(9,21)(11,23)(13,28)(17,29)(18,27);(2,8,7,20)(3,12)(4,17)(9,23,21,11)(10,24,22,30)(13,28)(14,15)(16,29)(18,27)(25,32,31,26)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=egi,f^c=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72260536, #72260647, #72260651, … |
| maximal normal subgroup of | #72260536, #72260647, #72260651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42707);