Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1033 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1033 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8)(5,12)(6,16)(9,17)(10,18)(11,14)(13,15);(5,13)(6,16)(8,17)(9,11)(12,15)(14,18);(3,4)(5,14)(6,9)(7,10)(8,12)(11,16)(13,18)(15,17)56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214886, D3×#12314, C2×#489347, … |
| maximal normal subgroup of | #72214886, C2×#489347, #72215816, … |
| maximal quotient of | C2×#143203, C2×#125647, C2×#125721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396070);