Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C16 · Schur multiplier (H2) = C211 |
| direct product | C23×C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C8 | C22 |
| commutator | C2 | C24×C16 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C25 | C2×C8 |
| prime radical | C26 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
5330 conjugacy classes of subgroups; only the 2114 normal subgroups are shown.
This group has 5330 conjugacy classes of subgroups — too many to draw. Only its 2114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4);(5,6);(7,8)(9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(8,10)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×#2112, C23×#2184, C23×#2187, … |
| maximal normal subgroup of | C23×#2112, C23×#2184, C23×#2187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529620);