Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C42 |
| direct product | C22×#2812 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
18898 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.
This group has 18898 conjugacy classes of subgroups — too many to draw. Only its 3490 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) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2);(3,4);(10,13)(14,16)(17,18)(19,22)(20,21)(23,24);(6,8)(10,13)(11,15)(18,21)(19,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#46350, C22×#40584, C22×#40484, … |
| maximal normal subgroup of | C22×#46350, C22×#40584, C22×#40484, … |
| maximal quotient of | C22×#16249, C2×#4414082, C22×#16255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540550);