Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7383 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D16 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C23 | #357 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,17)(14,26)(15,29)(16,21)(18,22)(19,23)(20,27)(24,36)(25,32)(28,42)(30,38)(31,37)(33,41)(34,39)(35,40);(12,17)(13,22)(14,23)(15,27)(18,19)(20,36)(21,32)(24,41)(25,39)(26,30)(28,40)(29,31)(33,42)(34,35)(37,38)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#150058, C2×#150057, C2×C8×D32, … |
| maximal normal subgroup of | C2×#150058, C2×#150057, C2×C8×D32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57250);