Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13667 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 783 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,9)(5,13)(6,16)(7,19)(10,22)(11,25)(12,26)(14,18)(15,27)(17,32)(20,31)(21,23)(24,33)(28,29)(30,34);(3,4)(5,14)(6,18)(7,21)(8,10)(9,13)(11,16)(12,19)(15,30)(17,33)(20,34)(22,27)(23,32)(24,31)(25,28)(26,29);(5,15)(9,22)(13,27)(14,30)(17,28)(23,26)(25,33)(29,32)62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#513022, C2×#513026, C2×#513025, … |
| maximal normal subgroup of | C2×#513022, C2×#513026, C2×#513025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420034);