Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 37748736 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C8 |
| direct product | C22×#1455 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2998 conjugacy classes of subgroups; only the 1922 normal subgroups are shown.
This group has 2998 conjugacy classes of subgroups — too many to draw. Only its 1922 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,7,10,8,9,11,12);(13,14,15,18,16,17,19,20);(6,9)(10,12)(14,17)(18,20)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#19745, #72214267, #72214336, … |
| maximal normal subgroup of | C22×#19745, #72214267, #72214336, … |
| maximal quotient of | C22×#7846, C2×D3×#1455 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249654);