Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C25×C8 |
| direct product | C82×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C82 | C22 |
| commutator | C2 | C22×C82 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C42 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1268 conjugacy classes of subgroups; only the 774 normal subgroups are shown.
This group has 1268 conjugacy classes of subgroups — too many to draw. Only its 774 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 — 4 generators on 20 points:
(1,2,3,4,5,6,7,8);(9,10,11,12,13,14,15,16);(17,18,19,20);(18,20)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C8×#1580, C2×C82×D4 |
| maximal normal subgroup of | C8×#1580, C2×C82×D4 |
| maximal quotient of | C8×#1580, C2×C82×D4, C8×#2171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91033);