Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12307 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| prime radical | C2×#3253 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
836 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 836 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);(3,4,6,9,7,12,17,11)(5,13,15,10,16,8,14,18);(4,8)(5,14)(6,17)(9,10)(11,18)(12,13)(15,16);(4,9,12,11)(8,10,13,18)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486701, C2×#486084, C2×#485585, … |
| maximal normal subgroup of | C2×#486701, C2×#486084, C2×#485585, … |
| maximal quotient of | #72265444, #72307458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392181);