Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C213×C4 |
| direct product | C22×#2273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C2 | C24×C42 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C27 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
24786 conjugacy classes of subgroups; only the 12322 normal subgroups are shown.
This group has 24786 conjugacy classes of subgroups — too many to draw. Only its 12322 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) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(9,10,12,11)(13,14)(15,19)(16,17)(18,20);(5,6,7,8)(9,11,12,10)(14,17)(18,19);(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,20)(17,19)272 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#34198, C22×#35390, C22×#34264, … |
| maximal normal subgroup of | C22×#34198, C22×#35390, C22×#34264, … |
| maximal quotient of | C22×C22⋊C42, C2×#513545, C22×#13794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532402);