Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C4×D16 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,8,14,10,17,32,16)(5,18,21,49,23,50,22,56)(6,15,25,41,26,12,9,33)(7,27,29,59,31,40,64,60)(11,35,37,55,39,54,47,20)(13,24,43,57,45,58,44,30)(19,51,52,66,53,48,34,42)(28,46,61,65,63,36,62,38);(3,5)(4,11)(6,19)(7,28)(8,21)(9,34)(10,23)(12,36)(13,42)(14,37)(15,46)(16,47)(17,39)(18,27)(20,30)(22,32)(24,35)(25,52)(26,53)(29,61)(31,63)(33,38)(40,50)(41,65)(43,51)(44,48)(45,66)(49,59)(54,58)(55,57)(56,60)(62,64);(3,6,8,25,10,26,32,9)(4,12,14,33,17,15,16,41)(5,19,21,52,23,53,22,34)(7,24,29,57,31,58,64,30)(11,36,37,38,39,46,47,65)(13,40,43,60,45,27,44,59)(18,48,49,42,50,51,56,66)(20,28,35,61,55,63,54,62)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512773, C2×#151086, C2×#153598, … |
| maximal normal subgroup of | C2×#512773, C2×#151086, C2×#153598, … |
| maximal quotient of | C2×#151080, C2×#151078, D3×#13656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419968);