Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7338 |
| commutator | C22×D4 | C2×C8 |
| Frattini, cosocle | #1783 | C22 |
| prime radical | #3484 | C4 |
| LCS | G ▷ C22×D4 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1783 ◁ G | |
| LECS | G ▷ #1783 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,10,6,11,14,16,7,12,15,8,5,9,3,13);(1,3)(2,8)(4,6)(5,15)(7,14)(9,13)(10,12)(11,16)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=dei,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72272684 |
| maximal normal subgroup of | #72272684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1720);