Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#2185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
4126 conjugacy classes of subgroups; only the 738 normal subgroups are shown.
This group has 4126 conjugacy classes of subgroups — too many to draw. Only its 738 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(5,6)(7,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17);(6,8)116 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#33520, C2×#499713, C22×#33831, … |
| maximal normal subgroup of | C22×#33520, C2×#499713, C22×#33831, … |
| maximal quotient of | C22×#12920 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529998);