Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13717 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2233 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#2218 | C2 |
| 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 | |
387 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 130 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)(11,27)(12,28)(14,29)(15,33)(17,23)(18,24)(20,32)(21,25)(26,34)(30,31);(5,14)(9,22)(13,29)(15,24)(17,30)(23,31)(25,28)(27,34);(3,4,7,21,8,10,19,12)(5,15,17,27,14,24,30,34)(6,18,16,11,20,33,32,26)(9,23,28,13,22,31,25,29)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, 420184);