Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 2062 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,11,13,12,17,9,15)(10,14,18,21,19,16,20,22);(3,4,5,6)(8,13)(9,11)(14,21)(15,17)(16,22)(18,20);(8,14)(9,11)(10,19)(13,21)(15,22)(16,17)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#494622, D3×#13055, #72261949, … |
| maximal normal subgroup of | C2×#494622, #72261949, #72261979, … |
| maximal quotient of | #72240576, #72240577, C2×#139787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401807);