Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C24×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C27 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
58050 conjugacy classes of subgroups; only the 31970 normal subgroups are shown.
This group has 58050 conjugacy classes of subgroups — too many to draw. Only its 31970 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(1,2);(3,4);(5,6);(7,8);(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(10,15)(14,23)(16,20)(17,24)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6437519, C2×#6437512, C2×#6437513, … |
| maximal normal subgroup of | C2×#6437519, C2×#6437512, C2×#6437513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494199);