Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#7426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1351 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
927 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 927 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,11)(6,14)(7,10)(8,12,17,15)(9,16)(13,18);(3,5)(4,8)(6,15,16,12)(7,13)(9,11,14,18)(10,17)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#150393, C2×#11710964, C2×#11710965, … |
| maximal normal subgroup of | C2×#150393, C2×#150394, #72284157, … |
| maximal quotient of | #72284157, #72291980, C2×#94701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57454);