Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#10316 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1604 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 1604 conjugacy classes of subgroups — too many to draw. Only its 410 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,12,9)(10,13,11,14)(16,20)(17,21)(18,22);(4,6)(8,13)(9,14)(10,11)(15,16,19,17)(18,21,22,20);(7,9,12,8)(10,14,11,13)(15,17,19,16)(18,20,22,21)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10316⋊C3, C2×#233527, #72214428, … |
| maximal normal subgroup of | C2×#10316⋊C3, C2×#233527, #72214428, … |
| maximal quotient of | C2×#121387, C2×#102869 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140555);