Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
| direct product | C4×#1412 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C2×C42 |
| Frattini, prime radical, cosocle | C2×#127 | C23 |
| socle | C22 | C2×#354 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,14,16,20)(8,12,18,11)(9,13)(10,15,19,17);(5,7,9,15)(6,10)(8,17,18,16)(11,14)(12,19)(13,20)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59129);