Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133875 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2270 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 2270 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,14,16,13)(12,15);(10,13,15,14);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=h,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fgi,f^c=fh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210012, #72210015, #72210018, … |
| maximal normal subgroup of | #72210012, #72210015, #72210018, … |
| maximal quotient of | #72210012, #72234190, C2×#134009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41205);