Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×#2199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4794 conjugacy classes of subgroups; only the 578 normal subgroups are shown.
This group has 4794 conjugacy classes of subgroups — too many to draw. Only its 578 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) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(8,16)(10,17)(13,19)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234427, C22×#33551, #72234569, … |
| maximal normal subgroup of | #72234427, C22×#33551, #72234569, … |
| maximal quotient of | C22×#13343, #72234536, C2×#508087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530054);