Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139842 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9499 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
600 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 600 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)(3,11)(4,16,19,31)(5,9,20,21)(6,10)(7,13,24,14)(8,18)(12,27)(15,23)(17,25)(22,30)(26,29,32,28);(1,3,5,14,6,15,20,13)(2,7,21,23,10,24,9,11)(4,12,18,29,19,30,17,28)(8,22,31,32,25,27,16,26);(1,4)(2,8)(3,12)(5,18)(6,19)(7,22)(9,16)(10,25)(11,26)(13,28)(14,29)(15,30)(17,20)(21,31)(23,32)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72255657, #72255830, #72310760, … |
| maximal normal subgroup of | #72255657, #72255830, #72310760, … |
| maximal quotient of | #72251663, #72251673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47038);