Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×#7099 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #106 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle, prime radical | C23 | #334 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
186 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(4,8,9,14)(5,11)(6,13)(10,17,16,15)(12,18)(19,20,21,24,22,25,27,30,23,26,28,31,29,32,33,34);(3,4,5,8,6,10,11,15,7,9,12,14,13,16,18,17)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#122921, #72227182, #72227197, … |
| maximal normal subgroup of | C2×#122921, #72227182, #72227197, … |
| maximal quotient of | #72227180, C2×#93877, C22×#7099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30114);