Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C22×#1619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
6738 conjugacy classes of subgroups; only the 778 normal subgroups are shown.
This group has 6738 conjugacy classes of subgroups — too many to draw. Only its 778 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20);(6,9)(7,12)(10,11)(14,18)(15,16)(19,20);(14,19)(15,16)(18,20)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#18524, C2×#209860, C2×#210060, … |
| maximal normal subgroup of | C22×#18524, C2×#209860, C2×#210060, … |
| maximal quotient of | #72215805, #72215816, C2×#488869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255213);