Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#13654 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C22×D32 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,6,5)(7,8,10,14)(9,13,20,26)(11,24,22,38)(12,25,23,17)(15,31,29,18)(16,33,30,19)(21,34,35,28)(27,37,32,36);(8,13)(9,18)(11,19)(12,23)(14,26)(15,16)(17,34)(20,31)(21,37)(22,33)(24,27)(25,28)(29,30)(32,38)(35,36);(3,5,6,4)(8,14)(13,26)(15,29)(16,30)(17,25)(24,38)(27,32)(28,34)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#2263, C2×#151085, C2×#513082, … |
| maximal normal subgroup of | C2×C4×#2263, C2×#151085, C2×#513082, … |
| maximal quotient of | C2×#151080, C22×#13654, D3×#13654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419956);