Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139806 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9497 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23903 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
765 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 765 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,6,12,7,10,5,11)(3,13,16,25,15,8,21,23)(4,17,20,24,18,26,19,9)(14,28,29,27,30,32,31,22);(1,3,6,21)(2,8,11,25)(4,14,19,29)(5,16,7,15)(9,22,26,32)(10,13,12,23)(17,28,24,27)(18,30,20,31);(1,4,7,18)(2,9)(3,14,15,30)(5,20,6,19)(8,22)(10,24)(11,26)(12,17)(13,27)(16,31,21,29)(23,28)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72237638, #72237640, #72302051, … |
| maximal normal subgroup of | #72237638, #72237640, #72302051, … |
| maximal quotient of | #72236931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47004);