Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124874 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1560 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7631 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 16 points:
(1,2)(3,10,11,16,12,15,13,6)(4,7)(5,9)(8,14);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14);(2,7,9,8)(6,15,16,10)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209722, #72272672, #72272673, … |
| maximal normal subgroup of | #72209722, #72272672, #72272673, … |
| maximal quotient of | #72227432, #72227433, #72227993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32070);