Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#7389 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1333 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 34 points:
(1,2);(4,9,10,24,12,26,13,27)(5,14,17,30)(6,19)(7,21)(11,15,20,31,23,32,29,25)(16,28)(18,34,33,22);(3,4)(5,15)(6,20)(7,10)(8,12)(9,22)(11,28)(13,21)(14,24)(16,29)(17,32)(18,25)(19,23)(26,34)(27,30)(31,33)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216279, #72216317, C2×AGL(1,ℤ/32), … |
| maximal normal subgroup of | #72216279, #72216317, C2×AGL(1,ℤ/32), … |
| maximal quotient of | D3×#7389, C4×#7389, C22×#7389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57266);