Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12173 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 138 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);(3,4)(5,9)(6,10)(7,12)(8,13)(11,14)(15,18)(16,17);(4,9)(5,14)(6,7)(10,11)(12,17)(13,18)(15,16);(4,10,13,12)(9,11,18,17)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486345, C2×#486084, C2×#485585, … |
| maximal normal subgroup of | C2×#486345, C2×#486084, C2×#485585, … |
| maximal quotient of | #72265117, #72307464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391850);