Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7316 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7316 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1996 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1996 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 18 points:
(1,2);(3,4,6,9)(5,11,12,17)(7,10,15,18)(8,14,16,13);(3,5)(4,8,10,11)(6,14,15,12)(7,13)(9,17)(16,18)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#139750, C2×#144364, #72219158, … |
| maximal normal subgroup of | C2×#139750, C2×#144364, #72219158, … |
| maximal quotient of | #72219414, #72225888, #72226347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35219);